Upload
science2010
View
227
Download
1
Embed Size (px)
Citation preview
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 1/344
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 2/344
1$785$/352'8
××××Q210$75,&(6
W. B. Vasantha KandasamFlorentin Smarandache
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 3/344
This book can be ordered from:
Zip Publishing1313 Chesapeake Ave.Columbus, Ohio 43212, USAToll Free: (614) 485-0721E-mail: [email protected] Website: www.zippublishing.com
Copyright 2012 by Zip Publishing and the Authors
Peer reviewers:Prof. Mihàly Bencze, Department of Mathematics
Áprily Lajos College, Braúov, RomaniaDr. Sebastian Nicolaescu, 2 Terrace Ave., West Orange, NJ 07052,Professor Paul P. Wang, Department of Electrical & Computer EngPratt School of Engineering, Duke University,Durham, NC 27708, USA
Many books can be downloaded from the followingDigital Library of Science:http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htm
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 4/344
&217(176
3UHIDFH
&KDSWHU2QH
,1752'8&7,21 &KDSWHU7ZR
32/<120,$/6:,7+0$75,;&2()),&,(17
&KDSWHU7KUHH
$/*(%5$,&6758&785(686,1*0$75,;
&2()),&,(1732/<120,$/6
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 5/344
&KDSWHU6L[
683(50$75,;/,1($5$/*(%5$6
&KDSWHU6HYHQ
$33/,&$7,2162)7+(6($/*(%5$,&6758&785(6:,7+1$785$/352'8&7
&KDSWHU(LJKW
68**(67('352%/(06
)857+(55($',1*
,1'(;
$%2877+($87+256
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 6/344
35()$&(
In this book the authors introduce a new product onatural product. We see when two row matrices multiplied, the product is taken component wise; fo
x2, x3, …, xn) and Y = (y1, y2, y3, … , yn) then X ×
…, xnyn) which is also the natural product of X wi
find the product of a n × 1 column matrix with anmatrix, infact the product is not defined. Thus if
X =
1
2
n
x
x
x
ª º« »
« »« »« »¬ ¼
# and Y =
1
2
n
y
y
y
ª º« »
« »« »« »¬ ¼
#
under natural product
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 7/344
this product is more natural which is just identica
replaced by multiplication on these matrices.Another fact about natural product is this enables th
two super matrices of same order and with same type see on supermatrices products cannot be defined easilyfrom having any nice algebraic structure on the col
matrices of same type.This book has eight chapters. The first chapter is
nature. Polynomials with matrix coefficients are introdtwo. Algebraic structures on these polynomials with mis defined and described in chapter three. Chapter
natural product on matrices. Natural product on suintroduced in chapter five. Super matrix linear algebrachapter six. Chapter seven claims only after this notion we can find interesting applications of them. The final
over 100 problems some of which are at research level.
We thank Dr. K.Kandasamy for proof reading and supportive.
:%9$6$1
)/25(17
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 8/344
&KDSWHU2QH
,1752'8&7,21
In this chapter we only indicate as referenc
concepts we are using in this book. Howevereader should refer them for a complete under
book.In this book we define the notion of nat
matrices so that we have a nice natural prod
column matrices, m × n (m ≠ n) matrices. This
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 9/344
The concept of polynomials with matrix coef
used. That is if p(x) = i
ii 0
a x∞
=
¦
where x is an indeterminate and if ai is a matrix (a sq
or a row matrix of a column matrix or a m × n mathen p(x) is a polynomial in the variable x wcoefficients (‘or’ used in the mutually exclusive sense
Suppose
p(x) =
32
0
1
ª º« »« »« »« »−¬ ¼
+
23
1
5
−ª º« »« »« »« »¬ ¼
x +
01
0
2
ª º« »« »« »« »¬ ¼
x3 +
70
1
0
ª º« »« »« »« »¬ ¼
x5
is a polynomial with column matrix coefficients.
We also introduce polynomial matrix coefficien
We call usual matrices as simple matrices.
The super matrix concepts are used. If X = (a1 a2 | a
∈ R (or Q or Z) then X is a super row matrix [8, 19].
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 10/344
then Y is a super column matrix.
Let
M =
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
x x x x
x x x x
x x x xx x x x
ª º« »« »« »« »« »¬ ¼
with xi ∈ R (or Q or Z); 1 ≤ i ≤ 16 be a super squ
P =1 4 7 10 13 16
2 5 8 11 14 17
3 6 9 12 15 18
a a a a a a
a a a a a a
a a a a a a
ª º« « « ¬ ¼
is a super row vector.
S =
1 2 3 4 5 6 7 8
9 10 11 16
17 18 19 24
25 26 27 32
a a a a a a a a
a a a ... ... ... ... a
a a a ... ... ... ... a
a a a ... ... ... ... a
ª º« »
« »« »« »¬ ¼
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 11/344
B =
1 2 3
4 5 6
7 8 9
10 11 12
13 14 15
16 17 18
19 20 21
a a a
a a aa a a
a a a
a a a
a a a
a a a
ª º
« »« »« »« »« »« »« »« »« »¬ ¼
with ai ∈ R (or Q or Z
is a super column vector [8, 19].
Also we use the notion of vector spaces, Svector spaces and Smarandache linear algebra [17].
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 12/344
&KDSWHU7ZR
32/<120,$/6:,7+0$75,;
&2()),&,(176
In this chapter we define polynomials in thecoefficients from the collection of matrices of scall such polynomials as matrix coefficient polynomials with matrix coefficients. We fexamples before we define operations on them.
Example 2.1: Let p(x) = (5, 3, 0, –3, 2) + (0, 10, 1, 0, 1)x2 + (–7, –9, 10, 0, 0)x5 – (3, 2, 1, 2, 1is a polynomial in the variable x with row matrix
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 13/344
Example 2.3: Let
p(x) =3 0
1 2
ª º« »−¬ ¼
+1 0
0 2
ª º« »¬ ¼
x2 +0 1
0 3
ª º« »¬ ¼
x3 +1
4
ª « ¬
1 4
0 0
ª º« »¬ ¼
x8 +0 0
1 2
ª º« »¬ ¼
x9 +0 1
5 0
ª º« »¬ ¼
x10
be a square matrix coefficient polynomial.
Example 2.4: Let
T(x) =2 10 1
5 2
ª º« »« »« »¬ ¼
+1 01 1
1 0
ª º« »« »« »¬ ¼
x +1 20 3
4 0
ª º« »« »« »¬ ¼
x3 +90
6
ª « « « ¬
be a polynomial with 3 × 2 matrix coefficient.
Now we define some operations on the collection.
DEFINITION 2.1: Let
V R =∞
=
-®
¯
¦ i
i
i 0
a x ai = (x1 ,…, xn)
are 1 × n row matrices, xi ∈ R (or Q or Z); 1 ≤ i
collection of all row matrix coefficient polynomia
group under addition.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 14/344
0 = (0,…,0) + (0,…,0)x + … + (0,… ,defined as the row matrix coefficient zero polyno
Let p(x) =∞
=
¦ i
i
i 0
a x now –p(x) =∞
=
−¦ i
i 0
a x
the inverse of the row matrix coefficient polyno
+) is an abelian group of infinite order.
Example 2.5: Let
VR =i
ii 0 a x
∞
=
-®̄¦ ai = (x1, x2, x3, x4) with x j ∈ Q
be the collection of row matrix coefficient polyngroup under addition.
For if p(x) = (0, 2, 1, 0) + (7, 0, 1, 2)x + (0, 1, 2, 0)x5 and
q(x) = (7, 8, 9, 10) + (3, 1, 0, 7)x + (3,0,1,4)x4 + (7, 1, 0, 0)x5 + (1, 2, 3, 4)x8 are in VR then
p(x) + q(x) = ((0, 2, 1, 0) + (7, 8, 9, 10)) + (1, 0, 7))x + ((1, 1, 1, 1) + (3, 0, 1, 4))x3 + ((0, 0,4))x4 + ((0, 1, 2, 0) + (7, 1, 0, 0))x5 + (1, 2, 3, 4)x
= (7 10 10 10) + (10 1 1 9)x + (4 1 2
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 15/344
Example 2.6: Let
VR = ii
i 0
a x∞
=
-®¯¦ ai = (x1, x2, x3); x j ∈ Z12; 1 ≤ j
be the collection of all row coefficient polynomialgroup under modulo addition 12.
Example 2.7: Let
VR =10
ii
i 0
a x=
-®¯¦ ai = (d1, d2) with d j ∈ Q; 1 ≤ j
be the row coefficient polynomial. VR is a gaddition.
Example 2.8: Let
VR =5
ii
i 0
a x=
-®¯¦ ai = (x1, x2); x1, x2 ∈ Z10}
be the row coefficient polynomial. VR is a finite gaddition.
We now can define other types of operations on V
We see if (1,1,1,1)x3 – (0, 0, 8, 27) = p(x) then ((0, 0, 2, 3)3 = p(x)
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 16/344
We know VR is a group under addition.
Now we can define product on VR as follows
Let p(x) = (0,1,2) + (3,4,0)x + (2,1,5)x2 + (3
q(x) = (6,0,2) + (0,1,4)x + (3,1,0)x2 + (1,2,3)
We define product of p(x) with q(x) as follow
p(x) × q(x)
= [(0,1,2) + (3,4,0)x+ (2,1,5)x2 + (3,0,2)x
(0,1,4)x + (3,1,0)x2
+ (1,2,3)x4
]= (0,1,2) (6,0,2) + (3,4,0) (6,0,2)x+(2,1
(3,0,2) (6,0,2)x3 + (0,1,2) (0,1,4)x + (3,(2,1,5) (0,1,4)x3 + (3,0,2) (0,1,4)x4 + (0(3,4,0) (3,1,0)x3 + (2,1,5) (3,1,0)x4 + (3
(0,1,2) (1,2,3)x
4
+ (3,4,0) (1,2,3)x
5
+ (2(3,0,2) (1,2,3)x7
= (0,0,4) + (18,0,0)x + (12,0,10)x2 + (18,0+ (0,4,0)x2 + (0,1,20)x3 + (0,0,8)x4
(9,4,0)x3 + (6,1,0)x4 + (9,0,0)x5 + (0,2,
+(2,2,15)x
6
+ (3,0,6)x
7
= (0,0,4) + (18,1,8)x + (12,5,10)x2 +
(6,3,14)x4 + (12,8,0)x5 + (2,2,15)x6 + (3,
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 17/344
Example 2.9: Let
VR =5
ii
i 0
a x=
-®¯¦ ai = (x1, …, x8); xi ∈ Q; 1 ≤ i
be a semigroup of row matrix polynomials. VR iunder product.
Now we see (VR, +, ×) is a commutativpolynomials with row matrix coefficients.
We give examples of them.
Example 2.10: Let
VR = {p(x) = ii
i 0
a x∞
=
¦ ; a j = (x1, x2…, x18); xi ∈ R; 1
be a ring of polynomials with row matrix coefficients
Now we have shown examples of polynomial coefficients in the variable x.
Example 2.11: Let
VC = ii
i 0
a x∞
=
-®¯¦ a j =
1
2
3
4
x
xx
x
ª º
« »« »« »« »« »« »¬ ¼
with xi ∈ Z; 1 ≤ i ≤
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 18/344
q (x) =
42
1
4
0
ª º« »« »« »« »−« »
« »¬ ¼
+
23
4
5
0
ª º« »« »« »« »« »« »¬ ¼
x +
23
1
4
5
ª º« »« »« »« »« »« »¬ ¼
x2 +
21
1
0
4
ª º« »−« »« »« »« »« »¬ ¼
x3 +
21
2
3
0
ª « −« « « « « ¬
p(x) + q(x) =
3
0
1
02
ª º« »« »« »
« »« »« »¬ ¼
+
4
2
1
40
ª º« »« »« »
« »−« »« »¬ ¼
+
1 2
0 3
x0 4
2 50 0
ª º ª º« » « »« » « »« » « »+
« » « »« » « »« » « »¬ ¼ ¬ ¼
+
+
2
1
1
0
4
ª º« »−
« »« »« »« »« »¬ ¼
x3 +
2
1
2
3
0
ª º« »−
« »« »« »« »« »¬ ¼
x4
=
7 3
2 3
x2 4
ª º ª º« » « »« » « »« » « »+ +« » « »
2
4
3
ª º« »« »« »« »
x2 +
2
1
1
ª º« »−« »« »« »
x3 +
2
1
2
ª « −« « «
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 19/344
Thus
VC = ii
i 0
a x∞
=
-®¯¦ ai =
1
2
n
x
x
x
ª º« »« »« »« »¬ ¼
#; xi ∈ Q (or R or Z) ; 1
is an abelian group under addition with polynomcoefficients are column matrices.
Now Vn×m denotes the collection of all polynomcoefficients are n×m matrices. Vn×m is a group under
Now if m ≠ n then on Vn×m we cannot define prwill illustrate this situation by an example.
Example 2.12: Let
V5×3 = ii
i 0
a x∞
=
-®¯¦ a j =
1 6 11
2 7 12
3 8 13
4 9 14
5 10 15
x x x
x x x
x x x
x x x
x x x
ª º
« »« »« »« »« »« »¬ ¼
where xi ∈ R;
be the group of polynomials under addition whose are 5×3 matrix.
Example 2.13: Let
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 20/344
Thus we can say
Vn×m = ii
i 0
a x∞
=
-®¯¦ ai =
11 12
21 22
n1 n2
a a ... aa a ... a
a a ... a
ª « « « « ¬
# #
is the group of polynomials in the variable x w
as n × m matrices. Clearly if n ≠ m we cannot dVn×m.
Now we can define product on Vn×n, that is wfirst illustrate this by an example.
Example 2.14: Let
Vn×n = ii
i 0
a x∞
=
-®¯¦ ai =
11 12
21 22
n1 n 2
a a ...
a a ... a
a a ... a
ª « « « « ¬
# #
where aij ∈ R; 1 ≤ i, j ≤ n}
be the group of polynomials under addition witmatrix coefficients. We see on Vn×n, one can
Vn×n is only a semigroup which is non commutativWe will illustrate this situation by examples.
Example 2.15: Let
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 21/344
We will show how addition in V3×3 is carried out.
Let p(x) =
0 3 2
1 0 0
0 0 4
−§ ·¨ ¸¨ ¸¨ ¸© ¹
+
2 1 0
3 0 2
1 2 3
§ ·¨ ¸¨ ¸¨ ¸© ¹
x2 +
0
1 2
2 1
§ ¨ ¨ ¨ ©
and
q(x) =
1 2 1
0 1 3
6 1 2
§ ·¨ ¸¨ ¸¨ ¸−© ¹
+
1 2 3
0 1 5
5 0 1
§ ·¨ ¸¨ ¸¨ ¸−© ¹
x +
1 2
2 3
3 2
−§ ¨
−¨ ¨ −©
0 1 0
9 0 1
0 2 3
§ ·¨ ¸¨ ¸¨ ¸© ¹
x3 be in V3×3.
p(x) + q(x) =0 3 21 0 0
0 0 4
−§ ·¨ ¸¨ ¸¨ ¸© ¹
+1 2 10 1 3
6 1 2
§ ·¨ ¸¨ ¸¨ ¸−© ¹
+10
5
§ ¨ ¨ ¨ −©
+
2 1 0 1 2 3
3 0 2 2 3 11 2 3 3 2 1
ª º−§ · § ·
« »¨ ¸ ¨ ¸+ −« »¨ ¸ ¨ ¸¨ ¸ ¨ ¸« »−© ¹ © ¹¬ ¼
x2
0 1 2 0 1 0ª º§ · § ·
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 22/344
1 3 3
1 3 32 4 4
§ ·¨ ¸¨ ¸¨ ¸−© ¹
x2
+
0 2 2
10 2 12 3 3
§ ·¨ ¸¨ ¸¨ ¸© ¹
x
We see V3×3 is an abelian group under additio
Example 2.16: Let
V2×2 = ii
i 0
a x∞
=
-®¯¦ ai = 1 2
3 4
x x
x x
§ ·¨ ¸© ¹
; xi ∈ R; 1
be the semigroup of polynomials in the vcoefficients from the collection of all 2 × 2 product.
p(x) =1 2
0 4
§ ·¨ ¸© ¹
+0 1
2 3
§ ·¨ ¸© ¹
x +1 2
3 0
§ ·¨ ¸© ¹
x
q(x) = 0 12 0
§ ·¨ ¸© ¹
+ 1 02 3
§ ·¨ ¸© ¹
x + 1 23 4
§ ·¨ ¸© ¹
x3 b
Now
p(x) . q(x) = 1 20 4
§ ·¨ ¸© ¹
0 12 0
§ ·¨ ¸© ¹
+ 0 12 3
§ ·¨ ¸© ¹
§ ¨ ©
1 2§ · 0 1§ · 2 1 2§ · 1 2§
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 23/344
+ 1 23 0§ ·¨ ¸© ¹
1 23 4§ ·¨ ¸© ¹
x5 + 4 18 0§ ·¨ ¸© ¹
+ 2 06 2§ ·¨ ¸© ¹
+4 1
0 3
§ ·¨ ¸© ¹
x2 +5 6
8 12
§ ·¨ ¸© ¹
x +2 3
8 9
§ ·¨ ¸© ¹
x2
+5 6
3 0
§ ·¨ ¸© ¹
x3 +7 10
12 16
§ ·¨ ¸© ¹
x3 +3 4
11 16
§ ·¨ ¸© ¹
x4 +7
3
§ ¨ ©
=4 1
8 0
§ ·¨ ¸© ¹
+7 6
14 14
§ ·¨ ¸© ¹
x +6 4
8 12
§ ·¨ ¸© ¹
x2 +12 1
15 1
§ ¨ ©
3 4
11 16
§ ·¨ ¸© ¹
x4 +7 10
3 6
§ ·¨ ¸© ¹
x5.
This is the way product is defined. Thus V2×2 is aunder multiplication.
V2×2 is a monoid and infact V2×2 has zero divisors
This is a polynomial ring.
Example 2.17: Let
V i∞-
®¦
11 12 13 14
21 22 23 24
a a a a
a a a a
§ ·¨ ¨
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 24/344
V4×4 is a group under addition and V4×4 is a s
product (V4×4, +, ×) is a ring which is non comring has zero divisors and units and all p(x) ofthan or equal to one have no inverse.
Example 2.18: Let
V2×2 = ii
i 0a x
∞
=
-®¯¦ ai = 11 12
21 22
a aa a
§ ·¨ ¸© ¹
;aij ∈ R; 1
be the ring of polynomials with 2×2 matrix covariable x. V2×2 is non commutative and has ze
no p(x) ∈V2×2, of degree greater than one hacannot have idempotent in them.
We can differentiate and integrate these pomatrix coefficients apart from finding roots in the
Now we first illustrate this situation by some
Example 2.19: Let
p(x) =3 0
1 2
ª º« »¬ ¼
+2 6
1 5
ª º« »¬ ¼
x +7 0
0 8
ª º« »¬ ¼
x2 –
+8 1
0 1
ª º« »¬ ¼
x4 –0 4
2 0
ª º« »−¬ ¼
x5
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 25/344
dp(x)dx = 0 + 2 61 5ª º« »¬ ¼
+ 2 7 00 8ª º« »¬ ¼
x – 3 3 10 0ª « ¬
+ 48 1
0 1
ª º« »¬ ¼
x3 – 50 4
2 0
ª º« »−¬ ¼
x4
=2 6
1 5
ª º« »¬ ¼
+14 0
0 16
ª º« »¬ ¼
x –9 3
0 0
ª º« »¬ ¼
x2
+32 4
0 4
ª º« »¬ ¼
x3 –0 20
10 0
ª º« »−¬ ¼
x4.
We seedp(x)
dxis again a matrix coefficient po
the variable x.
We can find the second derivative of p(x).
Consider
2d p(x)
dx
=14 0
0 16
ª º« »¬ ¼
– 29 3
0 0
ª º« »¬ ¼
x
+ 332 4
0 4
ª º« »¬ ¼
x2 – 40 20
10 0
ª º« »−¬ ¼
x3
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 26/344
Example 2.20: Let
V2×4 = ii
i 0
a x∞
=
-®¯¦ ai = 1 2 3
5 6 7
x x x
x x x
§ ¨ ©
where xi ∈ R; 1 ≤ i ≤ 8}
be the 2 × 4 matrix coefficient polynomial.
Let p(x) =1 0 2 4
0 3 0 5
§ ·¨ ¸© ¹
+3 1 5
0 4 0
§ ¨ ©
+ 3 4 2 40 0 0 3
−§ ·¨ ¸© ¹
x3 + 1 1 02 0 2
−§ ¨
−©
be a 2 × 4 matrix coefficient polynomial. To finof p(x).
dp(x)
dx=
3 1 5 2
0 4 0 5
§ ·¨ ¸© ¹
+ 33 4 2
0 0 0
−§ ¨ ©
+ 41 1 0 2
2 0 2 0
−§ ·¨ ¸
−© ¹
x3
=3 1 5 2
0 4 0 5
§ ·¨ ¸© ¹
+9 12 6 1
0 0 0 9
−§ ¨ ©
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 27/344
Consider
2
2
d p(x)
dx= 2
9 12 6 12
0 0 0 9
−§ ·¨ ¸© ¹
x + 34 4 0
8 0 8
−§ ¨
−©
=18 24 12 24
0 0 0 18
−§ ·¨ ¸
© ¹
x +12 12 0 2
24 0 24
−§ ¨
−©
We see2
2
d p(x)
dx ∈ V2×4.
If we consider the third derivative of p(x);
3
3
d p(x)
dx=
18 24 12 24
0 0 0 18
−§ ·¨ ¸© ¹
+
2
12 12 0 24
24 0 24 0
−§ ·
¨ ¸−© ¹ x
=18 24 0 24
0 0 0 18
−§ ·¨ ¸© ¹
+24 24 0 4
48 0 48 0
−§ ¨
−©
We see3
3
d p(x)dx
∈ V2×4.
Further the forth derivative.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 28/344
Example 2.21: Let
VR = ii
i 0
a x∞
=
-®¯¦ ai = (x1, …, x6); xi ∈ Z; 1
be a row matrix coefficient polynomial.
p(x) = (2,0,1,0,1,5) + (3,2,1,0,0,0)x + (0,1,0+ (0,–2,–3,0,0,0)x3 + (8,0,7,0,1,0)x5
To find the derivative of
p(x) = dp(x)dx
= 0 + (3,2,1,0,0,0) + 2(0,1,0,2,0,4)x + 3(0,–5(8,0,7,0,1,0)x4
= (3,2,1,0,0,0) + (0,2,0,4,0,8)x + (0,–6,(40,0,35,0,5,0)x4.
We seedp(x)
dxis in VR.
2
2
d p(x)
dx= (0,2,0,4,0,8) + 2 (0, –6,–9,0,0,0)
+ 4 (40,0,35,0,5,0)x3
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 29/344
Example 2.22: Let
VC = ii
i 0
a x∞
=
-®¯¦ ai =
1
2
3
4
x
x
x
x
ª º« »« »« »« »¬ ¼
where xi ∈ Q; 1 ≤ i
be a 4 × 1 column matrix coefficient polynomial.
Let p(x) =
2
0
4
0
ª º« »« »« »« »¬ ¼
+
3
2
1
4
ª º« »« »« »« »−¬ ¼
x +
0
1
2
3
ª º« »« »« »« »¬ ¼
x3 +
4
5
2
1
ª º« »« »« »« »¬ ¼
x6 belon
dp(x)
dx
=
3
2
14
ª º« »« »
« »« »−¬ ¼
+ 3
0
1
23
ª º« »« »
« »« »¬ ¼
x2 + 6
4
5
21
ª º« »« »
« »« »¬ ¼
x5
=
3
2
14
ª º« »« »
« »« »−¬ ¼
+
0
3
69
ª º« »« »
« »« »¬ ¼
x2 +
24
30
126
ª º« »« »
« »« »¬ ¼
x5 ∈ VC.
0ª º 24ª º
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 30/344
=
0
6
12
18
ª º« »« »« »« »¬ ¼
x +
120
150
60
30
ª º« »« »« »« »¬ ¼
x4 ∈ VC.
3
3
d p(x)
dx=
0
6
12
18
ª º« »« »« »« »¬ ¼
+ 4
120
150
60
30
ª º« »« »« »« »¬ ¼
x3
=
0
6
12
18
ª º« »« »« »« »¬ ¼
+
480
600
240
120
ª º« »« »« »« »¬ ¼
x3 ∈ VC.
4
4
d p(x)
dx= 3
480
600
240
120
ª º« »« »« »
« »¬ ¼
x2 =
1440
1800
720
360
ª º« »« »« »
« »¬ ¼
x2 ∈ VC.
Thus we see VC, Vm×n, Vn×n and VR are suderivative and all higher derivatives are in VC
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 31/344
Example 2.23: Let
p(x) =
3 0 1
5 6 0
1 0 8
ª º« »« »« »¬ ¼
+
0 2 1
6 1 0
1 2 6
ª º« »« »« »¬ ¼
x
+
8 0 0
0 7 0
0 0 11
ª º« »« »« »¬ ¼
x2 +
0 0 2
0 9 0
10 0 0
ª º« »« »« »¬ ¼
x3.
To integrate p(x) . ³ p(x)dx = 3 0 15 6 0
1 0 8
ª º« »« »« »¬ ¼
x + ½ 06
1
ª « « « ¬
1/3
8 0 0
0 7 00 0 11
ª º« »« »« »¬ ¼
x3
+ 1/4
0 0 2
0 9 010 0 0
ª º« »« »« »¬ ¼
x4
+
1 2 3
4 5 6
7 8 9
a a a
a a a
a a a
ª º« »« »« »¬ ¼
3 0 1
5 6 0
1 0 8
ª º« »« »« »¬ ¼
x +
0 1 1/ 2
3 1/ 2 0
1/ 2 1 3
ª « « « ¬
8/3 0 0
0 7 / 3 0
ª º« » 3
0 0 1/ 2
0 9 / 4 0
ª º« » 4
1aª «
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 32/344
To integrate p(x), ³ p(x)dx= (1,2,3,4,5)x + ½ (0,1,0,3,–1)x2 + 1/3(5,0,8
+ 1/4 (1,2,0,4,5)x4 + 1/5(–2,1,4,3,0)x5 + (ai ∈ Q; 1 ≤ i ≤ 5.
= (1,2,3,4,5) + (0,1/2,0,3/2, -1/2)x2 + (5/3,0+ (1/4,1/2,0,1,5/4)x4 + (–2/5, 1/5,4/5,3/5,0+ (a1,a2,a3,a4,a5).
Example 2.25: Let
p(x) =
3
0
12
4
5
ª º« »« »« »« »« »« »« »« »¬ ¼
+
0
1
20
4
8
ª º« »« »« »« »« »« »« »« »¬ ¼
x +
1
0
98
7
0
−ª º« »« »« »−« »« »« »« »« »¬ ¼
x3 +
7
8
910
3
7
ª º« »« »« »« »« »« »« »« »¬ ¼
x4 +
be a column matrix polynomial.
³ p(x)dx =
3
0
1
2
4
5
ª º« »« »« »
« »« »« »« »« »¬ ¼
x + 1/2
0
1
2
0
4
8
ª º« »« »« »
« »« »« »« »« »¬ ¼
x2
+ 1/4
1
0
9
8
7
0
−ª º« »« »« »−
« »« »« »« »« »¬ ¼
x4
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 33/344
=
3
0
1
2
45
ª º« »« »« »« »« »« »« »« »¬ ¼
x +
0
1/ 2
1
0
24
ª º« »« »« »« »« »« »« »« »¬ ¼
x2 +
1/ 4
0
9/ 4
2
7 / 40
−ª º« »« »« »−« »« »« »« »« »¬ ¼
x4 +
7 /5
8/ 5
9 /5
2
3/ 57 /5
ª º« »« »« »« »« »« »« »« »¬ ¼
x5 +
4 /3
1/ 3
2 /3
5/ 6
5/ 65/3
ª º« « « « « « « « ¬ ¼
Example 2.26: Let
p(x) =0 2 1 4
6 0 1 0
ª º« »¬ ¼
+3 6 2 9
0 2 1 7
ª º« »¬ ¼
x +0 2
2 0
ª « ¬
+2 1 0 0
0 0 1 2
ª º« »
¬ ¼
x4 +0 1 2 0
6 0 0 3
ª º« »
¬ ¼
x5.
We find the integral of p(x).
³ p(x)dx =0 2 1 4
6 0 1 0
ª º« »¬ ¼
x + 1/23 6 2
0 2 1
ª « ¬
+ 1/40 2 4 4
2 0 1 2
ª º« »¬ ¼
x4 + 1/52 1 0
0 0 1
ª « ¬
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 34/344
+0 1/ 2 1 1
1/ 2 0 1/ 4 1/ 2
ª º« »¬ ¼
x4 +2 /5 1/ 5
0 0
ª « ¬
+0 1/ 6 1/ 3 0
1 0 0 1/ 2
ª º« »¬ ¼
x6 + 1 2
5 6
a a
a a
ª « ¬
We see VC, VR, Vn×m and Vn×n under integrprovided the entries of the coefficient matrices tfrom Q or R. If they take the from Z they are nintegration only closed under differentiation.
We will illustrate this situation by a few exam
Example 2.27: Let
p(x) = (3, 8, 4, 0) + (2, 0, 4, 9)x + (1, 2, 1, 1)x3 + (3, 4, 8, 9)x5 where the coefficients matrices with entries from Z.
We find integral of p(x).³ p(x)dx = (3, 8, 4, 0)x + 1/2(2, 0, 4, 9)x2 + 1+ 1/4(1, 0, 1, 1)x4 + 1/6(3, 4, 8, 9)
We see (1, 0, 2, 9/4), (1/3, 2/3, 1/3, 1/3), (1(1/2, 2/3, 4/3, 3/2) ∉ Z × Z × Z × Z. Thus w
matrix coefficient polynomials with matrix entrnot closed under intervals that is ³ p(x)dx ∉ VC oVm×n if the entries are in Z.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 35/344
³ p(x) dx =
2
34
0
ª º« »
« »« »« »¬ ¼
x + 1/2
1
23
4
ª º« »
« »« »« »¬ ¼
x2 +1/3
0
01
1
ª º« »
« »« »« »¬ ¼
x3
+ 1/4
0
10
3
ª º
« »« »« »« »¬ ¼
x4 + 1/5
3
00
4
ª º
« »« »« »« »¬ ¼
x5 +
1
2
3
4
a
aa
a
ª º
« »« »« »« »¬ ¼
=
2
34
0
ª º
« »« »« »« »¬ ¼
x +
1/ 2
13/ 2
2
ª º
« »« »« »« »¬ ¼
x2 +
0
01/ 3
1/ 3
ª º
« »« »« »« »¬ ¼
x3 +
0
1/ 40
3/ 4
ª º
« »« »« »« »¬ ¼
x4
+
3/ 5
0
0
4 /5
ª º« »« »« »« »¬ ¼
x5 +
1
2
3
4
a
a
a
a
ª º« »« »« »« »¬ ¼
.
Clearly these column matrices do not take their Z.
Example 2.29: Let
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 36/344
³ p(x)dx =3 0 1 1
6 6 0 0
ª º« »¬ ¼ x + 1/2
1 2
0 1
ª « ¬
+ 1/33 0 0 1
0 2 2 0
ª º« »¬ ¼
x3 + 1/42 1 1
0 2 0
ª « ¬
+ 1/51 0 1 0
0 1 0 1
ª º« »¬ ¼
x5 + 1 2 3 4
5 6 7 8
a a a a
a a a a
ª º« »¬ ¼
ai
³ p(x)dx = 3 0 1 16 6 0 0ª º« »¬ ¼
x + 1/ 20 1ª « ¬
+1 0 0 1/ 3
0 2 /3 2/ 3 0
ª º« »¬ ¼
x3 +1/ 2 1/ 4
0 1/ 2
ª « ¬
+1/ 5 0 1/ 5 0
0 1/5 0 1/ 5
ª º« »¬ ¼
x5 + 1
5
a a
a a
ª « ¬
In view of this we have the following theorem
THEOREM 2.1: Let V R (or V C or V n× m or V m× mcoefficient polynomials with matrix entries from
Q The derivatives of every polynomial in V (o
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 37/344
COROLLARY 1: If in theorem, Z is replaced by Q
then every integral of the matrix coefficient polynom
(or V C or V n× m or V m× m).
Now we find or show some polynomial identicase of matrix coefficient polynomials.
Consider (1, 1, 1, 1, 1)x2 – (4, 9, 16, 25, 81) =
ii
i 0
a x∞
=
-®¯¦ a = (x1, x2, x3, x4, x5) with xi ∈ Z or Q or C
≤ 5}. Given (1, 1, 1, 1, 1)x2 – (4, 9, 16, 25, 81) = (0).
((1, 1, 1, 1, 1)x – (2, 3, 4, 5, 9)) ×((1, 1, 1, 1, 1)5, 9)) = (0)
Thus x = (2,3,4,5,9) or – (2,3,4,5,9).
Take the matrix coefficient polynomial(1,1,1)x3 – (27,8,125) = (0)
We can factorize (1,1,1)x3 – (27,8,125) = 0 as[(1,1,1)x – (3,2,5)] [(1,1,1)x2 + (3,2,5)x + (9,4
Take (1,1,1,1)x4 – (16,81,625,16) = (0)
We can factorize this polynomial as [(1(4,9,25,4)] [(1,1,1,1)x2 – (4,9,25,4)] = (0,0,0,0)
x2 = – (4,9,25,4)
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 38/344
Now we see yet another equation
p(x) = (1,1,1,1)x2 + (4,4,4,4)x + (4,4,4,4) = (
a matrix coefficient polynomial in the variable x
x =
2(4,4,4,4) (4,4,4,4) 4 (1,1,1,1)
(2,2,2,2)
r u
=(4,4,4,4) (0)
(2,2,2,2)
r
=(4,4,4,4)
(2,2,2,2)
= – (2,2,2,2).
Thus p(x) has coincident roots.
Consider (1,1,1)x3 – (6,3,9)x2 + (12,3,27)x +
= p(x) = (0,0,0) be a matrix coefficient polyn
To find the roots of p(x).
p(x) = (1,1,1)x3 – 3(2,1,3)x2 + 3(4,1,9)x –
= ((1,1,1)x – (2,1,3))3.
Thus x = (2,1,3), (2,1,3) and (2,1,3).
Now p(2,1,3) = (1,1,1) (2,1,3)3 – 3(2,1,3) (2,
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 39/344
Consider
1 0 2 1x
0 1 0 2
ª º§ · § ·−« »¨ ¸ ¨ ¸
© ¹ © ¹¬ ¼
1 0 3 7x
0 1 0 1
ª º§ · § ·+« »¨ ¸ ¨ ¸
© ¹ © ¹¬ ¼
Clearly p2 1
0 2
§ ·¨ ¸
© ¹
= (0) and p3 7
0 1
§ ·¨ ¸
© ¹
= (0).
We can consider any product of linear polynmatrix coefficients. However we see it is difficuequations in the matrix coefficients as even solving eusual polynomials is not an easy problem.
Now having seen the properties of matrix polynomials we now proceed onto discuss otherassociated with it.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 40/344
&KDSWHU7KUHH
$/*(%5$,&6758&785(686,1*
&2()),&,(1732/<120,$/6
In this chapter we introduce several typestructures on these matrix coefficient polynomthem.
Throughout this chapter VR denotes the colle
matrix coefficient polynomials. VR = ii
i 0
a x∞
=
-®¯¦
where yi ∈ R (or Q or C or Z); 1 ≤ ii d i }
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 41/344
Now Vn×m = ii
i 0
a x∞
=
-®¯¦ ak =
11 1m
21 2m
n1 nm
a ... a
a ... a
a ... a
§ ·¨ ¸¨ ¸¨ ¸¨ ¸© ¹
# # #aij ∈
Z or C); 1 ≤ i ≤ n; 1 ≤ j ≤ m} denotes the collection
matrix coefficient polynomial.
Finally Vn×n = ii
i 0
a x∞
=
-®¯¦ ak =
11 12
21 22
n1 n 2
a a ... a
a a ... a
a a ... a
§ ¨ ¨ ¨ ¨
©
# #
∈ R (or Q or Z or C); 1 ≤ i, j ≤ n} denotes the colln × n matrix coefficient polynomial.
We give algebraic structures on them.
THEOREM 3.1: V R , V C , V n× m and V n× n (m ≠ n) are graddition.
THEOREM 3.2: V R and V n× n are semigroups (mon
multiplication.
THEOREM 3.3: V R and V n× n are rings
(i) V R is a commutative ring.
(ii) Vn×n is a non commutative ring
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 42/344
We give examples of zero divisors.
Example 3.1: Let VR[x] = ii
i 0
a x∞
=
-®¯¦ ai = (x1, x2,
Q or R or Z, 1 ≤ j ≤ 5} be a matrix coefficient po
Take p(x) = (3,2,0,0,0) + (6,3,0,0,0)x
+ (7,0,0,0,0)x2 + (8,1,0,0,0)x4
q(x) = (0,0,1,2,3) + (0,0,0,4,2)x2 + (0,0+ (0,0,0,3,4)x4 + (0,0,0,5,2)x7 be elements in VR.
p(x) q(x) = (0,0,0,0,0).
Thus VR has zero divisors.
Considera(x) = (5,0,0,0,2) + (3,0,0,0,0)x + (0,0,0,0,7
+ (2,0,0,0,-1)x3 + (6,0,0,0,0)x5 and
b(x) = (0,1,2,3,0) + (0,0,1,2,0)x + (0,1,0,0,0+ (0,1,0,7,0)x3 + (0,2,0,4,0)x8 in VR.
We see (a(x)) × (b(x)) = (0,0,0,0,0). We see
constant polynomial certainly (q(x))2 ≠ q(x) forthen deg ((q(x). q(x)) = n2.
We show that VR and Vn×n have several non t
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 43/344
Example 3.3: Let Vn×n be a ring.
Take p(x) =3 16 2
§ ·¨ ¸© ¹
x3 +2 15 7
§ ·¨ ¸© ¹
x2 + 1 be in Vn×
Clearly ¢p(x)² generates a two sided ideal.
But p(x)Vn×n generates only one sided ideal. (Vn×n) (p(x)) is not a two sided ideal. Thus Vn×n
number of right ideals which are not left ideal andideals. Further Vn×n has left ideals which are not right
Example 3.4: Let
V4×4[x] = ii
i 0
a x∞
=
-®¯¦ ai = 1 2
3 4
x x
x x
§ ·¨ ¸© ¹
where x j ∈ Q;
be the matrix coefficient polynomial ring. Let
P = ii
i 0
a x∞
=
-®¯¦ ai = 1 2
3 4
x xx x
§ ·¨ ¸© ¹
where x j ∈ Z; 1 ≤ j ≤
P is only a subring of V4×4 and is not an ideal of V4×4.
THEOREM 3.6: Let V R and V n× n be matrix
polynomial rings. Both V R and V n× n have subringsnot ideals. We see if p(x) ∈ V R or V n× n; degree of p(x
of usual polynomials is the highest power of x whi
zero coefficient.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 44/344
p(x) =
3 1 2
0 1 50 0 1
§ ·¨ ¸
¨ ¸¨ ¸© ¹
+
7 2 1
0 5 76 1 2
§ ·¨ ¸
¨ ¸¨ ¸© ¹
x
2
+
2
00
§ ¨
¨ ¨ ©
+
2 1 5
6 7 8
0 1 2
§ ·¨ ¸¨ ¸
¨ ¸© ¹
x8.
The degree of p(x) is 8.
Now in case of usual polynomials if their from a field then every polynomial p(x) can be ma
However the same does not hold good in cand Vn×n.
Considerp(x) = (0,3,0,0)x4 + (1,2,3,4)x3 + (2,0,0,1)
VR. Clearly p(x) cannot be made into a
coefficient polynomial for (0,3,0,0) has no inverto multiplication.
Let q(x) = (5,7,8, –4)x5 + (1,2,3,0)x3 + (7,0,1be in VR. Now q(x) can be made as a monic matrpolynomial. For multiply q(x) by
t = (1/5, 1/7, 1/8, –1/4). Now tq(x) = (1,1,1,3/8, 0)x3 + (7/5,0,1/8, – 5/4)x + (8/5, 9/7, 0, –matrix coefficient polynomial of degree five.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 45/344
Clearly 3 01 0
§ ·¨ ¸© ¹
has no inverse or the matrix 31
§ ¨ ©
invertible.
Hence p(x) cannot be made into a monic matrix
polynomial in V2×2. Consider
p(x) =7 0
0 8
§ ·¨ ¸© ¹
x5 +1 8
7 5
§ ·¨ ¸© ¹
x4
+ 0 12 0
§ ·¨ ¸© ¹
x3 + 0 11 0
§ ·¨ ¸© ¹
x2 + 1 02 5
§ ·¨ ¸© ¹
in V2×
We see p(x) can be made into a monic polynomia
A =1/ 7 0
0 1/8
§ ·¨ ¸© ¹ is such that
1/ 7 0
0 1/8
§ ·¨ ¸© ¹
7 0
0 8
§ ·¨ ¸© ¹
=1 0
0 1
§ ·¨ ¸© ¹
.
Thus
1/ 7 0
0 1/8
§ ·¨ ¸© ¹
p(x) =1 0
0 1
§ ·¨ ¸© ¹
x5 +1/ 7 0
0 1/8
§ ·¨ ¸© ¹
1
7
§ ¨ ©
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 46/344
=0 1
1 0
§ ·¨ ¸
© ¹
x5 +1/ 7 8/ 7
7 /8 5/8
§ ·¨ ¸
© ¹
x4 +0
1/ 4
§ ¨
©
+0 1/ 7
1/8 0
§ ·¨ ¸© ¹
x2 +1/ 7 0
1/ 4 5/8
§ ·¨ ¸© ¹
has been made into a monic polynomial.
We have shown only some of the mapolynomials can be made and not all mapolynomials as the collection of row matrices n×n matrices are not field just a ring with zero div
Thus we have seen some of the propecoefficient polynomials. Unlike the number synot zero divisors, we cannot extend, all the rmatrix coefficients can also be zero divisors.
Thus we can say a matrix coefficient polyno
(or Vn×n) divides another matrix coefficient polVR (or Vn×n) if q(x) = p(x) b(x) where deg (b(x))deg p(x) < deg q(x).
We illustrate this by some examples. Suppose
p(x) = ((3,2,1) + (7,–1,9)x) ((1,1,2) + (1,1,1– (2,5,1)x) and
q(x) = ((7,–1,9)x + (3,2,1)) ((1,1,1)x + (1,1,(2 5 1) )) ((2 4 6) 2 (3 1 2) (1
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 47/344
Suppose X = (a1, a2, …, an) is a row matrix with ≤ n; we say X is a prime row vector or row matrix if
prime and none of the ai is zero. Thus (3,5,11,13)23,31) and (11,23,29,43,41,53,59,47,7,11) are matrices.
We say or define the row matrix (a1, …, an) divimatrix (b1, b2, …, bn) if none of the ai’s are zero fo
and ai /bi for every i, 1 ≤ i ≤ n. That is we say (a1, …, …, bn) if (b1 /a1, …, bn /an) = (c1, …, cn) and ci ∈Z(ai ≠ 0; i = 1, 2, …, n).
We will illustrate this situation by some examples
Let (5,7,2,8) = x and y = (10,14,8,8) we say x/y(10/5, 14/7, 8/2, 8/8) = (2,2,4,1).
Now if x = (0,2,3,5,7,8) and y = (5,4,6,10,21,24or y/x is not defined.
So when matrix coefficient polynomials are deavery very difficult to define division in VR.
Clearly if x = (a1, …, an) with ai ≠ 0 and ai primthen we see there does not exist any y = (b1, …, bn)
and bi ≠ 1 dividing x, (1 ≤ i ≤ n). Thus the only div(a1, …, an) are y = (1,1,…, 1) and y = (a1, a2, …, an) we face a lot of problems in dealing with matrix mand however we only multiply the two row matric
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 48/344
as follows; if x =
1
2
n
a
a
a
ª º« »
« »« »« »¬ ¼
#and y =
1
2
n
b
b
b
ª º« »
« »« »« »¬ ¼
#then the na
x with y denoted by x ×n y =
1
2
n
a
a
a
ª º« »« »« »« »¬ ¼
# ×n
1
2
n
b
b
b
ª º« »« »« »« »¬ ¼
# =
2
n
a
a
a
ª « « « « ¬
This product is defined as natural producolumn matrices and the natural product operatio
×n.
Example 3.5: Let x =
7
2
0
15
ª º« »« »« »« »« »« »¬ ¼
and y =
1
3
5
27
ª º« »« »« »« »« »« »¬ ¼
.
Now the natural product of x with y is x ×n y =
7 1º 7ª º
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 49/344
Now if x =
1
1
1
ª º« »
« »« »« »¬ ¼
#and y =
1
2
n
a
a
a
ª º« »
« »« »« »¬ ¼
#be any two n ×
matrices when x ×n y = y ×n x = y.
Thus x =
1
1
1
ª º« »« »« »« »¬ ¼
#acts as the natural product identi
infact any n × 1 collection of column vectors is aunder natural multiplication or natural product and i
and is a commutative monoid.
THEOREM 3.7: Let
V =
-ª º°« »
°« »®« »°« »°¬ ¼¯
#
1
2
n
a
a
a
ai ∈ Q (or Z or R); 1 ≤ i ≤ n}
be the collection of all n × 1 column matrice
commutative semigroup under natural pro
multiplication) of column matrices.
Proof is direct and hence is left as an exercise to t
E l 3 6 L t
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 50/344
We see x ×n y =
1
2
3
0
0
0
ª º« »
« »« »« »« »« »« »« »¬ ¼
×n
0
0
0
0
1
2
ª º« »
« »« »« »« »« »« »« »¬ ¼
=
0
0
0
0
0
0
ª º«
« « « « « « « ¬ ¼
Thus x is a zero divisor. Inview of thifollowing result.
THEOREM 3.8: Let
V =
-ª º°« »°« »®« »°« »°¬ ¼¯
#
1
2
n
a
a
a
ai ∈ Z (or Q or R); 1 ≤ i
be the semigroup under natural multiplicationdivisors.
This proof is also very simple.
Example 3.7: Let
V =
1
2
a
a
-ª º°« »°« »®« »#
ai ∈ Z; 1 ≤ i ≤ 6}
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 51/344
Take
W =
1
2
6
a
a
a
-ª º°« »°« »®« »°« »°¬ ¼¯
#ai ∈ 3Z; 1 ≤ i ≤ 6} ⊆ V;
W is a subsemigroup of V. Infact W is an ideal of the
V. Thus we have several ideals for V.
Example 3.8: Let
V =
1
2
10
a
a
a
-ª º°« »°« »®
« »°« »°¬ ¼¯
#ai ∈ Q; 1 ≤ i ≤ 10}
be the semigroup under natural product.
Consider
W =
1
2
10
aa
a
-ª º°« »°« »®« »°« »°¬ ¼¯
#ai ∈ Z; 1 ≤ i ≤ 10} ⊆
W is only a subsemigroup of V and is not an ideal of V
Take
a-ª º
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 52/344
From this example we see a subsemigroup ian ideal.
Inview of this we give the following resuwhich is simple.
THEOREM 3.9: Let
V =
-ª º°« »°« »®« »°« »°¬ ¼¯
#
1
2
n
a
a
a
ai ∈ Q (or R); 1 ≤ i ≤
be a semigroup under natural product. V has
which are not ideals. However every ideal is a s
Proof is left as an exercise for the reader.
Now we have the concept of Smarandache swill illustrate this situation by an example.
Example 3.9: Let
V =
1
2
8
a
a
a
-ª º
°« »°« »®« »°« »°¬ ¼¯
#ai ∈ Q; 1 ≤ i ≤ 8}
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 53/344
M is a subring as well as a group under natural produwe see M is not an ideal of V. Thus V is a S
semigroup.
Inview of this we can easily prove the following th
THEOREM 3.10: Let
V =
-ª º°« »°« »®« »°« »°¬ ¼¯
#
1
2
n
m
m
m
mi ∈ Q (or R); 1 ≤ i ≤ n}
be a semigroup under natural product. V is a Smsemigroup.
Proof: For take
M =
1
2
m
a
a
a
-ª º
°« »°« »®« »°« »°¬ ¼¯
#ai ∈ Q \ {0} or (ai ∈ R \ {0}); 1 ≤ i ≤
M is a group under natural product as every elemeinvertible, hence the theorem.
Now we proceed onto give an example or two.
Example 3 10: Let
C id h
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 54/344
Consider the set
P =1 1 1 11 , 1 , 1 , 1 ,
1 1 1 1
- − −ª º ª º ª º ª º°« » « » « » « »− −®« » « » « » « »°« » « » « » « »− −¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼¯
1 1 11 , 1 , 1 ,
1 1 1
− −ª º ª º ª º« » « » « »− « » « » « »« » « » « »− ¬ ¼ ¬ ¼ ¬ ¼
is a group under product. Thus Z is a Smarandac
Infact B =
1 1
1 , 1
1 1
- ½−ª º ª º° °« » « »−® ¾« » « »° °« » « »−¬ ¼ ¬ ¼¯ ¿
⊆ M is also a grou
Smarandache semigroup as B ⊆ P.
Now we wish to prove the following theorem
THEOREM 3.11: Let
M =
-ª º°« »°« »®« »°« »°¬ ¼¯
#
1
2
n
aa
a
ai ∈ Z (or Q or R); 1 ≤ i
be a semigroup under natural product. Smarandache subsemigroup then M is a
semigroup. However even if M is a Smaranda
every subsemigroup of M need not be a
W th th t f th th b
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 55/344
We prove the other part of the theorem by an exam
Consider
Y =
1
2
n
x
x
x
-ª º°« »°« »®« »°« »°¬ ¼¯
#xi ∈ Z; 1 ≤ i ≤ n}
be a semigroup under natural product.
Y is a Smarandache semigroup as
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 56/344
Y is a Smarandache semigroup as
P =
1 11 1
1 1
1 1,
1 1
1 1
1 1
1 1
- ½−ª º ª º° °« » « »−° °« » « »° °« » « »−° °« » « »
−° °« » « »® ¾« » « »−
° °« » « »° °−« » « »° °« » « »−° °« » « »° °−« » « »¬ ¼ ¬ ¼¯ ¿
⊆ Y
is a group under natural multiplication. Hesemigroup.
Take
W =
1
2
7
a
a
a
-ª º°« »°
« »®« »°« »°¬ ¼¯
# ai ∈ 3Z; 1 ≤ i ≤ 7} ⊆ Y
W is only a subsemigroup of Y and is not subsemigroup of Y. Hence even if Y is a S-semsubsemigroups whch are not Smarandache Hence the theorem.
Now we have seen ideals and subsemisubsemigroups about column matrix semigroup
DEFINITION 3 1: Let
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 57/344
DEFINITION 3.1: Let
M =
-ª º°« »°« »®« »°« »°¬ ¼¯
# # #
11 12 1n
21 22 2n
m1 m2 mn
a a ... a
a a ... a
a a ... a
aij ∈ Z (or Q or
1 ≤ i ≤ m and 1 ≤ j ≤ n}
be the collection of all m × n (m ≠ n) matrices. M un
multiplication / product × n is a semigroup.
If X =
ª º
« »« »« »« »¬ ¼
# # #
11 12 1n
21 22 2n
m1 m2 mn
a a ... a
a a ... a
a a ... a
and Y =
ª
« « « « ¬
# #
11 12
21 22
m1 m2
b b
b b
b b
be any two m × n matrices in M.
We define X × m Y =
ª « « « «
¬
# #
11 11 12 12 1n
21 21 22 22 2n
m1 m1 m2 m2 mn
a b a b ... a
a b a b ... a
a b a b ... a
Clearly X × m Y is in M. (M, × n) is defined as the
under natural product.
be any two 3 × 5 matrices We find the natur
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 58/344
be any two 3 × 5 matrices. We find the naturwith Y.
X ×n Y =
2 1 0 5 1
0 3 1 2 5
1 4 3 0 1
ª º« »« »« »−¬ ¼
×n
3 2 0
4 0 1
0 1 2
ª « « « ¬
=6 2 0 5 30 0 1 10 35
0 4 6 0 5
ª º« »« »« »¬ ¼
.
Example 3.12: Let
S =
1 2 3
4 5 6
28 29 30
a a a
a a a
a a a
-ª º°« »°« »®« »°« »°¬ ¼¯
# # #ai ∈ Z; 1 ≤ i ≤
be the semigroup under natural product. S issemigroup with identity. S has infinite numbesubsemigroups which are not ideals.
Example 3.13: Let
1 2a a
a a
-ª º°« »°« »
Take
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 59/344
Take
I =
1 2
3 4
a a
0 0
0 0
0 0
a a
-ª º
°« »°« »°« »°
« »®« »°« »°« »°
« »°¬ ¼¯
# #ai ∈ Q; 1 ≤ i ≤ 4}
It is easily verified I is an ideal of P under natu×n.
Consider the subsemigroup
S =
1 2
3 4
5 6
a a
a a
0 0
0 0
a a
-ª º°« »°« »°« »°
« »®
« »°« »°« »°« »°¬ ¼¯
# #ai ∈ Q; 1 ≤ i ≤ 6} ⊆ P
under natural product. Clearly S is an ideal of P.
Suppose
T =
1 2
3 4
a a
a a
-ª º°« »°« »®« »# #
ai ∈ Z; 1 ≤ i ≤ 12} ⊆ P
our natural product we are able to define some
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 60/344
our natural product we are able to define some on column matrices and rectangular matrices. N
onto define natural product on usual square matri
Let A = (aij)n×n and B = (bij)n×n be square mat(or Q or R); 1 ≤ i, j ≤ n. We define the natural pr
A ×n B as A ×n B = (aij)n×n (bij)n×n
= (aij bij)n×n = (cij)n×n.
We will illustrate this by few examples.
Example 3.14: Let
A =
6 1 2
0 3 4
2 1 0
§ ·¨ ¸¨ ¸¨ ¸© ¹
and B =
3 0 1
2 1 0
0 1 2
§ ¨ ¨ ¨ ©
be two 3 × 3 matrices. To find the natural produc
A ×n B =
6 1 2
0 3 4
2 1 0
§ ·¨ ¸¨ ¸¨ ¸© ¹
3 0 1
2 1 0
0 1 2
§ ·¨ ¸¨ ¸¨ ¸© ¹
=
18
0
0
§ ¨ ¨ ¨ ©
Now the usual matrix product of A with B is
6 1 2§ ·¨ ¸
3 0 1§ ·¨ ¸
We see A.B ≠ A ×n B in general. Further
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 61/344
We see A.B ≠ A ×n B in general. Further operation ‘.’ the usual matrix multiplication is non c
where as the natural product ×n is commutative.
We just consider the following examples.
Example 3.15: Let
M = 3 42 0
ª º« »¬ ¼
and N = 1 20 1
ª º« »¬ ¼
be any two 2 × 2 matrices.
M.N = 3 42 0
ª º« »¬ ¼
1 20 1
ª º« »¬ ¼
= 3 102 4
ª º« »¬ ¼
and N.M =1 2
0 1
ª º« »¬ ¼
3 4
2 0
ª º« »¬ ¼
=7 4
2 0
ª º« »¬ ¼
.
We see M.N ≠ N.M.
However M ×n N =3 4
2 0
ª º« »¬ ¼
×n 1 2
0 1
ª º« »¬ ¼
=3 8
0 0
ª º« »¬ ¼
and
Example 3.16: Let
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 62/344
p
M =
7 0 0 00 8 0 0
0 0 2 0
0 0 0 4
ª º« »« »« »« »¬ ¼
and N =
1 0 00 2 0
0 0 3
0 0 0
ª « « « « ¬
We find M.N =
7 0 0 00 8 0 0
0 0 2 0
0 0 0 4
ª º« »« »« »« »¬ ¼
1 00 2
0 0
0 0
ª « « « « ¬
=
7 0 0 0
0 16 0 0
0 0 6 0
0 0 0 16
ª º« »« »« »« »¬ ¼
.
Also
N.M =
1 0 0 0
0 2 0 0
0 0 3 0
0 0 0 4
ª º« »« »« »« »
¬ ¼
7 0 0 0
0 8 0 0
0 0 2 0
0 0 0 4
ª º« »« »« »« »
¬ ¼
=
7
0
0
0
ª « « « «
¬
7 0 0 0ª º 1ª
7 0 0 0ª º« »
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 63/344
=
0 16 0 0
0 0 6 0
0 0 0 16
ª º« »
« »« »« »¬ ¼
. We see M.N = M ×n N
In view of this we have the following theorem.
THEOREM 3.12: Let
M =
-ª º°« »°« »°« »®
« »°« »°« »°¬ ¼¯
# # # # #
1
2
3
n
a 0 0 0 ... 0
0 a 0 0 ... 0
0 0 a 0 ... 0
0 0 0 0 ... a
ai ∈ Q (or Z or
1 ≤ i ≤ n}
be the collection of all n × n diagonal matricesemigroup under natural product and M is also a
under usual product of matrices and both the ope
identical on M.
Proof: Let
1
2
a 0 0 0 ... 0
0 a 0 0 ... 0
ª º« »« »
1b 0 0 0 ... 0ª º« »
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 64/344
and
1
2
3
4
n
0 b 0 0 ... 0
0 0 b 0 ... 0
0 0 0 b 0
...
0 0 0 0 ... b
« »
« »« »« »« »« »« »« »¬ ¼
# # # # #
be two matri
Now let us consider the natural product of
A ×n B =
1 1
2 2
3 3
4 4
a b 0 0 0 ...
0 a b 0 0 ...
0 0 a b 0 ...
0 0 0 a b
...
0 0 0 0 ...
ª «
« « « « « « « ¬
# # # #
Consider the matrix product;
A.B =
1 1
2 2
3 3
4 4
a b 0 0 0 ...
0 a b 0 0 ...
0 0 a b 0 ...0 0 0 a b
...
0 0 0 0
ª « « « « « « « «¬
# # # #
multiplication. Both the operations on M are
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 65/344
general.
The proof is direct and hence left as an exerreader.
THEOREM 3.14: Let
M = {(aij) | aij ∈ Z (or Q or R or C); 1 ≤ i, j ≤
be a semigroup under natural product. M is a Smsemigroup.
Proof: Let P = {(aij) | aij ∈ Z \ {0}, (R] {0} or Q]{0}) 1 ≤ i ≤ n} ⊆ M be a group under natural muSo M is a S-semigroup.
It is pertinent to mention here that these semigideals subsemigroups, zero divisors and idempotenSmarandache analogue.
Now we proceed onto give more structures
product.
DEFINITION 3.2: Let
M =
-ª º°« »
°« »®« »°« »°¬ ¼¯
#
1
2
m
a
a
a
ai ∈ Q (or Z or R or C); 1 ≤ i ≤
Example 3.17: Let
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 66/344
M =
1
2
3
4
aa
a
a
-ª º°« »°« »®« »°« »°¬ ¼¯
ai ∈ R (or Q or Z); 1 ≤ i
be a ring under + and ×n. The reader can eaA ×n (B+C) = A ×n B + A ×n C where A, B acolumn matrices.
Consider A =
1
2
n
a
a
a
ª º« »« »« »« »¬ ¼
# , B =
1
2
n
b
b
b
ª º« »« »« »« »¬ ¼
# and C =
A ×n (B+C) =
1
2
n
a
a
a
ª º« »
« »« »« »¬ ¼
# ×n
1 1
2 2
n n
b c
b c
b c
§ ª º ª ¨ « » «
¨ « » « +¨ « » « ¨ « » « ¨ ¬ ¼ ¬ ©
# #
=
1
2
n
a
a
a
ª º
« »« »« »« »¬ ¼
# ×n
1 1
2 2
n n
b c
b c
b c
+ª º
« »+« »« »« »
+¬ ¼
#=
1 1 1
2 2 2
n n n
a (b c
a (b c
a (b c
+ª
« +« « «
+¬
#
Now consider A ×n B + A ×n C
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 67/344
=
1
2
n
aa
a
ª º« »« »« »« »¬ ¼
# ×n
1
2
n
bb
b
ª º« »« »« »« »¬ ¼
#+
1
2
n
aa
a
ª º« »« »« »« »¬ ¼
# ×n
1
2
n
cc
c
ª º« »« »« »« »¬ ¼
#
=
1 1 1 1
2 2 2 2
n n n n
a b a c
a b a c
a b a c
ª º ª º« » « »« » « »+« » « »« » « »¬ ¼ ¬ ¼
# #=
1 1 1 1
2 2 2 2
n n n n
a b a c
a b a c
a b a c
+ª º« »+« »« »« »
+¬ ¼
#.
Thus we see ×n distributes over addition. Now collection of all m × n matrices (m ≠ n) with entriesZ or Q or C or R. We see this collection also uaddition and natural product is a ring. Let
M = {(aij)m×n | m ≠ n; aij ∈ R (or Z or Q or C1 ≤ i ≤ m and 1 ≤ j ≤ n};
M is a ring infact a commutative ring.
However M is not a ring under matrix addition
product.
Example 3.18: Let
1 1ª º« »
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 68/344
M is a commutative ring with unit
1 1
1 1
1 1
« »
« »« »« »¬ ¼
.
M has units, zero divisors, subrings and ideal
Take a =
3 45 8
1 9
4 7
ª º« »« »« »« »¬ ¼
and b =
1/ 3 1/ 41/ 5 1/8
1 1/ 9
1/ 4 1/ 7
ª « « « « ¬
clearly ab = ba =
1 1
1 1
1 1
1 1
ª º« »« »« »« »¬ ¼
.
Consider a = 1 2
3 4
0 0
a a
a a
0 0
ª º« »« »« »« »¬ ¼
and b =
1 2
3 4
a a
0 0
0 0
a a
ª « « « « ¬
0 0
0 0
ª º« »
Take
-ª º
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 69/344
P =
1
2
3
4
a 0
a 0
a 0
a 0
-ª º
°« »°« »®« »°« »°¬ ¼¯
ai ∈ Q; 1 ≤ i ≤ 4} ⊆ M;
P is an ideal of M.
Consider
T =
1 2
3 4
5 6
7 8
a a
a a
a a
a a
-ª º°« »°« »®« »°« »°¬ ¼¯
ai ∈ Z; 1 ≤ i ≤ 8} ⊆ M;
clearly T is only a subring of M and is not an ideal oM has subrings which are not ideals. We can subrings which are not ideals.
Example 3.19: Let
N = 1 2 3
4 5 6
a a a
a a a
-ª º°®« »
¬ ¼°̄ai ∈ Z; 1 ≤ i ≤ 6}
be the ring under matrix addition and natural produM has no units.
Example 3 20: Let
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 70/344
Example 3.20: Let
M =
1 2 3
4 5 6
31 32 33
a a a
a a a
a a a
-ª º°« »°« »®« »°« »°¬ ¼¯
# # #ai ∈ Q; 1 ≤ i ≤
be a ring under matrix addition and natural prodring.
For consider
P =
1 2 3
4 5 6
31 32 33
b b b
b b b
b b b
-ª º°
« »°« »®« »°« »°¬ ¼¯
# # #bi ∈ 3Z; 1 ≤ i ≤
P is not an ideal of M. M has units, zero diviso
ideals.
Take
W =
1 2 3
4 5 6
a a a
0 0 0
0 0 0
a a a
-ª º°« »°« »
°« »®« »°« »°« »°¬ ¼¯
# # # ai ∈ Q; 1 ≤ i ≤ 6
Consider
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 71/344
S =
1 2 3a a a0 0 0
0 0 0
-ª º°« »°« »®« »°« »°¬ ¼¯
# # #ai ∈ Z; 1 ≤ i ≤ 3} ⊆
is only a subring and not an ideal.
Example 3.21: Let
M =
1 2
3 4
21 22
a a
a a
a a
-ª º°« »°« »®« »°« »°¬ ¼¯
# # ai ∈ Q; 1 ≤ i ≤ 22}
be a ring M is commutative ring with
1 1
1 1
1 1
ª º« »« »
« »« »¬ ¼
# #
a
respect to natural multiplication. M is not an integM has zero divisors and every element M is torsion fr
For consider x =
1 2
3 4
a a
a aª º« »« »« »« »
# # ∈ M.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 72/344
Example 3.24: Let
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 73/344
P =
1 2 3
4 5 6
31 32 33
a a aa a a
a a a
-ª º°« »°« »®« »°« »°¬ ¼¯
# # #ai ∈ Z; 1 ≤ i ≤ 33
be a ring P has no units. P has ideals. P has subringnot ideals. P has no idempotents or nilpotents.
Every element x in P is such that for no n ∈ Z+,
xn =
1 1 1
1 1 1
1 1 1
ª º« »« »« »« »¬ ¼
# # #.
THEOREM 3.15: Let
M =
-ª º°« »°« »®« »°« »°¬ ¼¯
# # #
11 1n
21 2n
m1 mn
a ... a
a ... a
a ... a
aij ∈ Q; 1 ≤ i ≤ m; 1 ≤
be a ring M is a S-ring.
COROLLARY 2: Every matrix ring under natu
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 74/344
y g
S-ring.
If Q is replaced by Z in the theorem and comatrix ring is not a S-ring.
Example 3.25: Let
S =
1 2 3
4 5 6
7 8 9
10 11 12
a a a
a a a
a a a
a a a
-ª º°« »°« »®« »°« »°¬ ¼¯
ai ∈ Z; 1 ≤ i ≤
be a ring, S is a S-ring.
Example 3.26: Let
M =
1 2
3 4
5 6
7 8
a a
a aa a
a a
-ª º°
« »°« »®« »°« »°¬ ¼¯
ai ∈ Q; 1 ≤ i ≤ 8
be a ring. M is a S-ring. For M has 8 subfields g
F1 =
1a 00 0
0 0
-ª º°« »°« »®« »°
a1 ∈ Q} ⊆ M is
0 0
a 0
-ª º°« »°
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 75/344
F3 =1a 0
0 0
0 0
°« »®« »°« »°¬ ¼¯
a1 ∈ Q} ⊆ M is a fie
F4 = 1
0 0
0 a0 0
0 0
-ª º°« »°« »®« »°« »°¬ ¼¯
a1 ∈ Q} ⊆ M is a fie
F5 =1
0 0
0 0
a 0
0 0
-ª º
°« »°« »®« »°« »°¬ ¼¯
a1 ∈ Q} ⊆ M is a fie
F6 =1
0 00 0
0 a
0 0
-ª º°« »°« »®« »°« »°¬ ¼¯
a1 ∈ Q} ⊆ M is a fie
F7 =
0 0
0 0
0 0
-ª º°« »°« »®« »°
a1 ∈ Q} ⊆ M is a field
1a 0-ª º
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 76/344
N =
1
20 a0 0
0 0
ª º
°« »°« »®« »°« »°¬ ¼¯
a1, a2 ∈ Q} ⊆
N is a subring and an ideal and not a field. Thufields. M is a S-ring. N also is a S-subring
subrings of M are not S-subrings.
For consider
S =
1 2
3 4
5 6
7 8
a a
a aa a
a a
-ª º°« »°« »®« »°« »°¬ ¼¯
ai ∈ Z; 1 ≤ i ≤ 8}
S is only a subring and clearly S is not a S-sub
has infinite number of subrings which are not S-su
Example 3.27: Let
W =1 2 3
4 5 6
7 8 9
a a a
a a a
a a a
-ª º
°« »®« »°« »¬ ¼¯
ai ∈ Q; 1 ≤ i ≤
b i d l l i li i f i
Further we can get compatability of natural producolumn and rectangular matrices. Hence we see the
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 77/344
under natural product can serve better purpose for behave like the real numbers or complex number or integers on which they are built. Now we give mostructure on them. Consider the set of all row ma{(x1, …, xn) | xi ∈ R+ ∪ {0} (or Q+ ∪ {0} or Z+ ∪ {n}, M under + is a commutative semigroup with (0,
its additive identity.
M under ×n is also semigroup. Thus (M, +semiring. We see this semiring is a commutative sezero divisors.
Suppose
S = {(x1, …, xn) | xi ∈ R+ ∪ {0} (or Z+ or Q+); 1
Now {S ∪ {(0, 0, …, 0)} = T, +, ×n} is a semifield.
It is easily verified T has no zero divisors and strict semiring for a = (x1, x2, …, xn) and b = (y1, ysuch that x+y = 0 implies a = (0) = b = (0, 0, …, 0will give examples of them before we proceed ontodescribe more properties.
Example 3.28: Let
M = {(a1, a2, a3) where ai ∈ Z+ ∪ {0}; 1 ≤ i ≤ 3}; (
Example 3.29: Let
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 78/344
T =
1
2
3
4
5
6
a
a
a
a
aa
-ª º°« »°« »°« »°
« »®« »°
« »°« »°« »°¬ ¼¯
ai ∈ Q+ ∪ {0}; 1 ≤ i ≤
be a semiring under + and ×n.
We see if x =
03
1
0
2
5
ª º« »« »« »« »« »« »
« »« »¬ ¼
and y =
10
0
2
0
0
ª º« »« »« »« »« »« »
« »« »¬ ¼
are in T
x ×n y =
0
3
1
0
2
ª º« »
« »« »« »« »« »
×n
1
0
0
2
0
ª º« »
« »« »« »« »« »
=
0
0
0
0
0
ª º« »
« »« »« »« »« »
.
Example 3.30: Let
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 79/344
M =
1 2 3
4 5 6
13 14 15
a a aa a a
a a a
-ª º°« »°« »®« »°« »°¬ ¼¯
# # #ai ∈ R+ ∪ {0}; 1 ≤ i ≤
be a semiring under + and ×n. Clearly M is commuta strict semiring. However M does contain zero divi
=
1 2 3
4 5 6
a a a
0 0 0
0 0 0
0 0 0
a a a
ª º« »« »« »
« »« »« »¬ ¼
and N =1 2 3
4 5 6
7 8 9
0 0 0
a a a
a a a
a a a
0 0 0
ª º« »« »« »
« »« »« »¬ ¼
with ai ∈ R
in M then
T ×n N =
1 2 3
4 5 6
a a a
0 0 0
0 0 0
0 0 0
a a a
ª º
« »« »« »« »« »« »¬ ¼
×n 1 2 3
4 5 6
7 8 9
0 0 0
a a a
a a a
a a a
0 0 0
ª º
« »« »« »« »« »« »¬ ¼
=
0
0
0
0
0
ª
« « « « « « ¬
Thus M is not a semifield.
Example 3.31: Let
For take a =1
2 3
0 0 a
0 a a
ª º« »« » and b =
1 2
3
b b
b 0
ª « «
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 80/344
4 5 6a a a« »« »¬ ¼ 0 0
« « ¬
we see a.b =1
2 3
4 5 6
0 0 a
0 a a
a a a
ª º« »« »« »¬ ¼
1 2
3
b b 0
b 0 0
0 0 0
ª º« »« »« »¬ ¼
=
Thus J is only a strict commutative semirinsemifield, we show how we can build semifields.
First we will illustrate this situation by some Example 3.32: Let
M = {(0,0,0,0), (x1, x2, x3, x4) | xi ∈ Q+; 1 ≤ i ≤
be a semifield. For we see (M, +) is a commutwith additive identity (0,0,0,0).
Further (M, ×n) is a commutative semigrouas its multiplicative identity.
Also M is a strict semiring for (a,b,c,d) + (x,y
= (a + x, b + y, c + z, t + d)
Example 3.33: Let
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 81/344
M =
1
2
9
10
aa
a
a
-ª º°« »°« »°« »®
« »°« »°« »°¬ ¼¯
# where ai ∈ R+, 1 ≤ i ≤ 10} an
P = M ∪
0
0
00
- ½ª º° °« »° °« »° °« »® ¾
« »° °
« »° °« »° °¬ ¼¯ ¿
# ; (P, +, ×n) is a semifield.
Example 3.34: Let
S =
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
17 18 19 20
a a a a
a a a a
a a a a
a a a a
a a a a
-ª º°« »°« »°« »®
« »°« »°« »
°¬ ¼¯
where ai ∈ R+, 1 ≤ i
0 0 0 0- ½ª º° °« »
Example 3.35: Let
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 82/344
T = 1 2
3 4
a aa a
-ª º°®« »¬ ¼°̄
ai ∈ Z+, 1 ≤ i ≤ 4} a
P = T ∪ 0 0
0 0
- ½ª º° °® ¾« »° °¬ ¼¯ ¿
; (P, +, ×n) is a sem
We see by defining natural product on minfinite number of semifields apart from R+ ∪
and Z+ ∪ {0}. We proceed onto give examples semirings. Recall a semiring S is a Smarandach
contains a proper subset T such that T under theis a semifield.
Example 3.36: LetM = {(a1, a2, …, a10) | ai ∈ Q+ ∪ {0}, 1 ≤
be a semiring under + and ×n. Take
T = {(0,a,0,…,0) | a ∈ Q+ ∪ {0}} ⊆T is a subsemiring of M. T is strict and T has nso T is a semifield under + and ×n. Hence M issemiring.
Example 3.37: Let1
2
a
a
-ª º°« »°« »
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 83/344
Consider
P =
0
0
0
a
-ª º°« »°« »°« »®« »°
« »°« »°¬ ¼¯
# a ∈ Z+ ∪ {0}} ⊆ T.
P is a subsemiring of T which is strict and has no zeThus T is a Smarandache semiring.
Example 3.38: Let
V =1 2 3 4
5 6 7 8
9 10 11 12
a a a a
a a a a
a a a a
-ª º°« »®« »°« »¬ ¼¯
where ai ∈ Z+ ∪ {0}, 1
be a semiring under + and ×n.
V is a Smarandache semiring as
a 0 0 0-ª º°« »
Example 3.39: Let
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 84/344
M =
1 2 3 4 5
6 7 8 9 10
11 12 13 14 15
16 17 18 19 20
21 22 23 24 25
a a a a aa a a a a
a a a a a
a a a a a
a a a a a
-ª º°« »°« »°« »®
« »°« »°« »°¬ ¼¯
where ai ∈ R+ ∪ {0}, 1 ≤ i ≤ 25}
be a semiring under + and ×n.
Consider
S =1
0 0 0 0 0
0 0 a 0 0
0 0 0 0 0
0 0 0 0 00 0 0 0 0
-ª º°« »°« »°« »®
« »°« »°« »°¬ ¼¯
a ∈ Z+ ∪ {0}
is a semiring as well as a semifield under + and ×a Smarandache semiring.
We can now define subsemirings andsubsemirings. These definitions are a matter hence left as an exercise to the reader. We h
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 85/344
Consider
X =1 2 3 4a a a a
0 0 0 0
0 0 0 0
-ª º°« »®« »°« »
¬ ¼¯
a ∈ 3Z+ ∪ {0}; 1 ≤ i ≤
x under + and ×n is a subsemiring of P. HoweverSmarandache subsemiring.
But we see P is a Smarandache semiring for
V =
d 0 0 0
0 0 0 0
0 0 0 0
-ª º°« »®« »°« »¬ ¼¯
d ∈ Z+ ∪ {0}} ⊆ P
is a semiring as well as a semifield under + and ×n.
Example 3.41: Let
P =
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
a a a a
a a a aa a a a
a a a a
-ª º°« »°« »®« »°« »°¬ ¼¯
where ai ∈ Q+ ∪ {0}, 1
P is a Smarandache semiring for take
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 86/344
W =
a 0 0 00 0 0 0
0 0 0 0
0 0 0 0
-ª º°« »°« »®« »°« »°¬ ¼¯
a ∈ Z+ ∪ {0}}
is a semifield under + and ×n. Hence P is
semiring. However P has infinitely many subsare not Smarandache subsemirings.
Consider
Pn =
1 2 3a a a 0
0 0 0 0
0 0 0 0
0 0 0 0
-ª º°« »°« »®« »°« »°¬ ¼¯
a ∈ nZ; 1 ≤ i ≤ 3; n
Pn is a subsemiring of P but is not a Smarandacof P. Thus we see in general all subsemirings ofsemiring need not be a Smarandache subsemiringsemiring which has a Smarandache subsemiring Smarandache semiring.
Inview of this we have the following theorem.
THEOREM 3.16: Let S, be a semiring of n × m
entries from R+∪ {0} (or Q
+∪ {0} or Z
+∪ {
and X ⊆ P so X ⊆ P ⊆ S that is X is a proper subsetis a semifield, so S is a Smarandache semiring.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 87/344
To show every subsemiring of a Smarandachneed not be a Smarandache subsemiring, we give an e
Consider
P = 1 2 3 4
5 6 7 8
a a a aa a a a
-ª º°®« »°¬ ¼¯
where ai ∈ Z+ ∪ {0}, 1
be a semiring under + and ×n.
If is easily verified P is a Smarandache semiring a
X =a 0 0 0
0 0 0 0
-§ ·°®¨ ¸
© ¹°̄a ∈ Z+ ∪ {0}} ⊆ P
is a semifield under + and ×n; so P is a Smarandache s
Consider a subsemiring
T = 1 2 3 4
5 6 7 8
a a a a
a a a a
-ª º°
®« »°¬ ¼¯where ai ∈ 5Z+ ∪ {0}, 1 ≤
clearly T is a subsemiring of P; however T is not a S
Example 3.42: Let
1 2 3-ª º°« »
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 88/344
P =
1 2 3
4 5 6
7 8 9
10 11 12
13 14 15
a a aa a a
a a a
a a a
a a a
-ª º°« »°« »°« »®
« »°« »°« »°¬ ¼¯
where ai ∈ Z+ ∪ {0},
be a semiring and + and ×n.
To show M has zero divisors.
Consider x =1 2 3
4 5 6
0 0 0
a a a
0 0 0
a a a
0 0 0
ª º« »« »« »« »« »« »¬ ¼
and y =
1 2
4 5
7 8
a a
0 0
a a
0 0
a a
ª « « « « « « ¬
We see x ×n y =
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
ª º« »« »« »« »« »« »¬ ¼
.
Thus M has zero divisors. Infact M has
b
1 2
0 0 0
b b 0
0 0 0
ª º« »« »« » h b b 3Z i M
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 89/344
b = 0 0 00 0 0
0 0 0
« »« »« »« »¬ ¼
where b1, b2 ∈ 3Z+ in M
we see a.b =
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
ª º« »« »« »« »« »« »¬ ¼
.
Thus we can get any number of zero divisors in M
M has no idempotents other than elements of the f
x =
1 1 1
0 0 01 1 1
0 0 0
0 0 0
ª º
« »« »« »« »« »« »¬ ¼
∈ M, we see x2 = x or
0 0 0
1 0 0
ª º« »« »« » i h h 2 d
The proof is direct hence left as an exercise to
We call all these idempotents only as trit d id t t t f thi th
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 90/344
generated idempotents; apart from this these mwith natural product do not contain any other idem
Example 3.43: Let M = {(a, b) | a, b ∈ Z+ ∪ {0under + and ×n. The only trivial idempotents of M
1), (1, 0) and (1, 1).
Example 3.44: Let P =a b
c d
-ª º°®« »
¬ ¼°̄a, b, c, d ∈
semiring under + and ×n.
The trivial idempotents of P are
0 0 1 0 0 1 0 0 0 0, , , , ,
0 0 0 0 0 0 1 0 0 1
-ª º ª º ª º ª º ª º ª°®« » « » « » « » « » «°¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼ ¬¯
0 0 0 1 1 1 1 0, , , ,
1 1 0 1 1 1 1 1
ª º ª º ª º ª º« » « » « » « »¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼
0 1 1 1 1 1 1 0 0 1, , , ,
1 1 1 0 0 1 0 1 1 0
½ª º ª º ª º ª º ª º°¾« » « » « » « » « »°¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼¿
we see I under natural product ×n is a semigroupl d d
The proof involves only simple numbertechniques, hence left as an exercise to the reader.
Example 3 45: L t
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 91/344
Example 3.45: Let
M = 1 2 3 4
5 6 7 8
a a a a
a a a a
-ª º°®« »°¬ ¼¯
where ai ∈ Z+ ∪ {0}, 1
be a semiring under + and ×n.
I =0 0 0 0 1 0 0 0 0 1 0 0
, ,0 0 0 0 0 0 0 0 0 0 0 0
-ª º ª º ª °®« » « » « °¬ ¼ ¬ ¼ ¬ ¯
0 0 1 0 0 0 0 1,
0 0 0 0 0 0 0 0
ª º ª º« » « »¬ ¼ ¬ ¼
,
0 0 0 0 0 0 0 0 0 0 0 0 0 0, , ,
1 0 0 0 0 1 0 0 0 0 1 0 0 0
ª º ª º ª º ª « » « » « » «
¬ ¼ ¬ ¼ ¬ ¼ ¬
1 1 0 0 0 1 1 0 0 0 1 1 1 0, , ,
0 0 0 0 0 0 0 0 0 0 0 0 1 0
ª º ª º ª º ª « » « » « » « ¬ ¼ ¬ ¼ ¬ ¼ ¬
1 1 1 1 1 1 1 1,1 1 1 0 1 1 1 1 ½ª º ª º°¾« » « »
°¬ ¼ ¬ ¼¿ ⊆ M;
x =1 1 0 0
0 0 0 0
ª º« »¬ ¼
and y =0 0 1 1
1 1 1 1
ª º« ¬ ¼
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 92/344
x ×n y =0 0 0 0
0 0 0 0
ª º« »¬ ¼
.
Each element in I \
1 1 1 1
1 1 1 1
- ½ª º° °
® ¾« »° °¬ ¼¯ ¿ can genthe semigroup.
For consider x =1 1 1 1
0 0 0 0
ª º« »¬ ¼
∈
¢x² =
0 0 0 0 1 1 1 1 0 1 0 1, , ,
0 0 0 0 0 0 0 0 0 0 0 0
-ª º ª º ª º ª°®« » « » « » «°¬ ¼ ¬ ¼ ¬ ¼ ¬¯
1 0 0 0 0 0 1 0 0 0 1 1, , ,
0 0 0 0 0 0 0 0 0 0 0 0
ª º ª º ª º ª« » « » « » «¬ ¼ ¬ ¼ ¬ ¼ ¬
1 1 0 0 0 1 1 0 1 0 1 0
, , ,0 0 0 0 0 0 0 0 0 0 0 0
ª º ª º ª º ª
« » « » « » «¬ ¼ ¬ ¼ ¬ ¼ ¬
1 1 1 0 0 1 1 1 1 1 0 1 1ª º ª º ª º ª
0 0 0 0 1 0 0 0 0 0 0 0 0, , ,
0 0 0 0 0 0 0 0 1 0 0 0 0
-ª º ª º ª º ª°= ®« » « » « » «
°¬ ¼ ¬ ¼ ¬ ¼ ¬¯
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 93/344
0 0 0 0 0 1 0 0 0 0 0 0 1 1, , ,
0 0 1 0 0 0 0 0 0 1 0 0 0 0
ª º ª º ª º ª « » « » « » « ¬ ¼ ¬ ¼ ¬ ¼ ¬
0 1 1 0 1 0 1 0 0 0 0 0 0 0
, , ,0 0 0 0 0 0 0 0 1 1 0 0 0 1
ª º ª º ª º ª
« » « » « » « ¬ ¼ ¬ ¼ ¬ ¼ ¬
0 0 0 0 1 0 0 0 0 1 0 0 0 0, , ,
1 0 1 0 1 0 0 0 0 1 0 0 0 0
ª º ª º ª º ª « » « » « » « ¬ ¼ ¬ ¼ ¬ ¼ ¬
0 0 1 0 1 0 0 0 0 1 0 0 0 1, , ,
0 0 1 0 0 1 0 0 1 0 0 0 0 0
ª º ª º ª º ª « » « » « » « ¬ ¼ ¬ ¼ ¬ ¼ ¬
1 0 0 0 0 0 1 0 0 0 1 0, ,
0 0 1 0 1 0 0 0 0 1 0 0
ª º ª º ª º« » « » «
¬ ¼ ¬ ¼ ¬ ¼and so on}.
We see order J is 26. Thus every singleton othand identity generate an ideal in the trivial
semigroup.Infact {0} generates {0} the trivial zero
1 1 1 1ª º
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 94/344
&KDSWHU)RXU
1$785$/352'8&7210$75,&
In this chapter we construct semivectosemifields and vector spaces over fields using of matrices under natural product.
DEFINITION 4.1: Let V be the collection of all
with entries from Q (or R) or C. (V, +) is an ab
is a vector space over Q (or R) according as V
from Q (or R). If V takes its entries from Q; V
space over R however if V takes its entries from
space over Q as well as vector spaces over R. W
spaces V (m ≠ n) are also linear algebras for us
product we get the linear algebra.
Example 4.2: Let
1 2a a-ª º°« »
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 95/344
M =3 4
5 6
7 8
9 10
a aa a
a a
a a
a a
-ª º°« »°« »°« »®
« »°« »°« »°¬ ¼¯
ai ∈ R; 1 ≤ i ≤ 10}
be a vector space over Q. Clearly M is of infinite dim
Example 4.3: Let
P = 1 2
3 4a aa a
-ª º°®« »¬ ¼°̄
ai ∈ R; 1 ≤ i ≤ 4}
be a vector space over R. Clearly dimension of P ove
Example 4.4: Let
M =
1
2
20
aa
a
-ª º°« »°« »®« »°« »°¬ ¼¯
#ai ∈ Q; 1 ≤ i ≤ 20}
be a vector space over R. Clearly M is not a vector
R. Clearly dimension of M over Q is 20.
Example 4.5: Let
The concept of subspace is a matter of routileft as an exercise to the reader.
However we give examples of them.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 96/344
However we give examples of them.
Example 4.6: Let
M =
1 2
3 4
5 6
a a
a aa a
-ª º°« »®« »°« »¬ ¼¯
ai ∈ Q; 1 ≤ i ≤ 6
be a vector space over Q.
Consider
T =1
2
3
a 0
0 a
a 0
-ª º°« »®« »°« »¬ ¼¯
ai ∈ Q; 1 ≤ i ≤ 3} ⊆
it is easily verified T is a subspace of M over Q.
Consider
P =1
2
3
a 0
a 0
a 0
-ª º°« »®« »
°« »¬ ¼¯
ai ∈ Q; 1 ≤ i ≤ 3} ⊆
P is also a subspace of M over Q. Now we con
Example 4.7: Let
P1 2 3a a a
-ª º°« »
® Q 1 ≤ i ≤ 9}
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 97/344
P = 4 5 6
7 8 9
a a aa a a
a a a
ª º°« »®« »°« »¬ ¼¯
ai ∈ Q; 1 ≤ i ≤ 9}
be a vector space Q.
Consider
M1 =1 2
3
a a 0
0 0 0
0 0 a
-ª º°« »®« »°« »
¬ ¼¯
ai ∈ Q; 1 ≤ i ≤ 3} ⊆
M1 is a subspace of P over Q.
Consider
M2 =
1
2
0 0 a
0 a 00 0 0
-ª º°« »
®« »°« »¬ ¼¯a1, a2 ∈ Q} ⊆ P
is also a subspace of P over Q.
However we see M1 ∩ M2 =
0 0 0
0 0 0
0 0 0
ª º« »« »« »¬ ¼
.
Take
0 0 0-ª º°
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 98/344
M4 =
1
0 0 0
0 0 0
a 0 0
-ª º°« »®« »°« »¬ ¼¯
ai ∈ Q} ⊆ P
is also a subspace of P over Q.
We see P = M1 + M2 + M3 + M4 and Mi ∩ M
if i ≠ j. Thus we can write P as a direct sum of su
Example 4.8: Let
P =
1
2
12
a
a
a
-ª º°« »°« »®« »°« »°¬ ¼¯
#ai ∈ Q; 1 ≤ i ≤ 12}
be a vector space over Q.
Consider
X1 =
1
2
aa
0
-ª º°« »°« »°« »®
« »a1, a2 ∈ Q} ⊆ P,
1
2
0
a
a
-ª º°« »°« »°« »°« »°
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 99/344
X2 = 3a
0
0
°« »°®« »°« »°« »°« »°« »
¬ ¼°̄
#
ai ∈ Q; 1 ≤ i ≤ 3} ⊆ P
is again a subspace of P over Q.
Take
X3 =
1
2
3
4
00
a
a
a
a
0
0
0
00
-ª º°« »°« »°« »°« »°« »°« »
°« »°®« »°« »°« »°« »°« »°« »
°« »°« »¬ ¼°̄
ai ∈ Q; 1 ≤ i ≤ 4} ⊆ P
Consider
0
00
-ª º°« »°« »°« »
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 100/344
X4 =1
2
3
4
5
0
0
0
a
a
a
a
a
°« »°« »°« »°« »°« »®
« »°« »°« »°« »°« »°« »°« »°¬ ¼¯
ai ∈ Q; 1 ≤ i ≤ 5} ⊆ P
is a subspace of P over Q.
We see Xi ∩ X j ≠
0
0
0
00
ª º« »« »« »« »« »
« »« »« »¬ ¼
#if i ≠ j.
Thus P is not a direct sum. However we see
P ⊆ X1 + X2 + X3 + X4, thus we say P is
direct sum of subspaces of P over Q.
Thus we have seen examples of direct sudirect sum of subspaces Interested reader can su
matrix product ×n. This is the vital difference and imdefining natural product ×n of matrices of same order
Now we define special strong Smarandache vecto
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 101/344
DEFINITION 4.2: Let
M =
-ª º°
« »®« »°« »¬ ¼¯
#
1
n
a
a
ai∈ Q; 1 ≤ i ≤ n}.
We define M as a natural Smarandache speci
characteristic zero under usual addition of matric
natural product × n. Thus {M, +, × n } is natural Smspecial field.
We give an example or two.
Example 4.9: Let
V =
1
2
7
aa
a
-ª º°« »°« »®« »°« »°¬ ¼¯
#ai ∈ Q; 1 ≤ i ≤ 7}
is a natural Smarandache special field of characteristi
Consider
Th
1
1
11
ª º« »« »
« »« »« » t th lti li ti id tit
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 102/344
Thus 1
1
1
1
« »« »« »« »« »« »¬ ¼
acts as the multiplicative identity.
Example 4.10: Consider the collection of matrices V with entries from Q.
We see
a
0
0
-ª º
°« »°« »®« »°« »°¬ ¼¯
#a ∈ Q} = P is a proper subset
field hence natural S-special field.
Example 4.11: Let
M =
1
2
3
4
5
a
a
a
a
a
-ª º°« »°« »°« »®
« »°« »°« »°¬ ¼¯
ai ∈ Q; 1 ≤ i ≤ 5}
Example 4.12: Let V = {(x1, x2, x3, x4, x5) | xi ∈ Rbe the special real natural Smarandache special fieldmatrices of characteristic zero.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 103/344
Example 4.13: Let V = {(x1, x2) | xi ∈ R, 1 ≤ i ≤ 2row matrix natural Smarandache special field of chzero.
All these fields are non prime natural Smarandafields for they have several natural S-special subfields
Now we can define the natural Smarandache spem × n matrices (m ≠ n).
Let
V =
11 12 1n
21 22 2n
m1 m2 mn
a a ... a
a a ... a
a a ... a
-ª º°« »°« »®« »°« »°¬ ¼¯
# # #aij ∈ R, 1 ≤ i ≤ m, 1
V is the special m × n matrix of natural Smspecial field of characteristic zero.
Example 4.14: Let
a a a a-ª º
Example 4.15: Let
1 2a a-ª º°« »
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 104/344
M = 3 4
21 22
a a
a a
°« »°« »®« »°« »°¬ ¼¯
# #ai ∈ R, 1 ≤ i ≤ 2
be the 11 × 2 matrix of natural Smarandache characteristic zero.
Example 4.16: Let
M = 1 2 16
17 18 32
a a ... aa a ... a
-ª º°®« »
¬ ¼°̄ai ∈ R, 1 ≤
be the 2 × 16 matrix of natural Smarandache spec
Now having seen natural S-special fields of (m ≠ n). We now proceed onto define the nospecial Smarandache field of square matrices.
Let
P =
11 12 1n
21 22 2n
a a ... a
a a ... a
-ª º°« »°« »® aij ∈ R 1 ≤ i
Example 4.17: Let
M =1 2 3
4 5 6
a a a
a a a
-ª º°« »®« »°
ai ∈ R, 1 ≤ i ≤ 9}
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 105/344
7 8 9a a a« »
°« »¬ ¼¯
be the 3 × 3 square matrix of natural special Smarand
Example 4.18: Let
M =
1 2 3 4 5
6 7 8 9 10
21 22 23 24 25
a a a a a
a a a a a
a a a a a
-ª º°« »°« »®« »°« »°¬ ¼¯
# # # # #ai ∈ R, 1 ≤ i
be the 5 × 5 square matrix of natural special Smaranof characteristic zero.
Now having seen and defined the concept of maspecial Smarandache field we are in a position to deSmarandache special strong matrix vector spaces.
DEFINITION 4.3: Let
V = {(x1 , x2 , …, xn) | xi ∈ Q (or R), 1 ≤ i ≤
be an additive abelian group.
row vector space over the natural row matrix
special field F R.
First we proceed onto give a few examples of
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 106/344
Example 4.19: Let M = {(x1, x2, x3, x4) | xi ∈ QSmarandache special strong row vector space oSmarandache special row matrix field
FR = {(x1, x2, x3, x4) | xi ∈ Q; 1 ≤ i ≤
Example 4.20: Let P = {(x1, x2, x3, …, x10) | xi ∈be a Smarandache special strong row vector natural special row matrix Smarandache field
FR = {(x1, x2, …, x10) | xi ∈ Q; 1 ≤ i ≤
Example 4.21: Let T = {(x1, x2, x3, …, x7) | xi ∈be a Smarandache special strong row vector natural special row matrix Smarandache field
FR = {(x1, x2, …, x7) | xi ∈ Q; 1 ≤ i ≤
Now we proceed onto define natural Scolumn matrix vector space over the special natural S-special field FC.
DEFINITION 4.4: Let
-ª º
F C =
1
2
a
a
a
-ª º°« »°« »
®« »°« »°¬ ¼¯
#
ai ∈ Q (or R), 1 ≤ i ≤ n}
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 107/344
na°¬ ¼¯
be the special column matrix natural S-field. Clear
vector space over the natural Smarandache special f
define V as a S-special strong column matrix vectorthe special column matrix natural S-field F C .
We will illustrate this situation by some examples
Example 4.22: Let
V =
1
2
10
x
x
x
-ª º°« »°« »®« »°« »°¬ ¼¯
#xi ∈ Q, 1 ≤ i ≤ 10}
is a S-special strong column matrix vector spacspecial column matrix natural S-field
FC =
1
2
x
x
-ª º
°« »°« »®« »°« »°¬ ¼
#xi ∈ Q (or R), 1 ≤ i ≤ 10}.
Example 4.23: Let
1
2
a
a
-ª º°« »°« »°« »
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 108/344
V = 3
4
5
a
a
a
°« »®« »°« »°« »°¬ ¼¯
ai ∈ R; 1 ≤ i ≤ 5}
be a S-special strong column matrix vector special column matrix natural S-field
FC =
1
2
3
4
5
x
xx
x
x
-ª º°« »°« »°« »®
« »°« »°« »°¬ ¼¯
xi ∈ R; 1 ≤ i ≤ 5}.
Example 4.24: Let
V = 1
2
x
x
-ª º°®« »
¬ ¼°̄xi ∈ R; 1 ≤ i ≤ 2}
be a S-special strong column matrix vector special column matrix natural S-field
Let
M =
11 12 1n
21 22 2n
a a ... a
a a ... a
-ª º°« »°« »®« »°« »
# # #aij ∈ Q (or R
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 109/344
m1 m2 mna a ... a°« »°¬ ¼¯
1 ≤ i ≤ m, 1 ≤ j ≤ n; m ≠ n}
be a group under matrix addition. Define
Fm×n (m≠n) =
11 12 1n
21 22 2n
m1 m2 mn
a a ... a
a a ... a
a a ... a
-ª º°« »°« »®
« »°« »°¬ ¼¯# # #
aij ∈ Q (or R), 1 ≤ i ≤ m, 1 ≤ j ≤ n}
to be special m × n matrix natural S-special field. N
M is a vector space over Fm×n called the S-special stmatrix vector space over the special m × n matrixspecial field Fm×n.
We will illustrate this situation by an example or
Example 4.25: Let
1 2 3a a a-ª º°
F6×3 =
1 2 3
4 5 6
a a a
a a a
-ª º°« »°« »®« »°« »
# # #ai ∈ Q; 1 ≤ i
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 110/344
16 17 18a a a°« »°¬ ¼¯
Example 4.26: Let
V =1 2 3 4 5 6 7
9 10 11 12 13 14 15
17 18 19 20 21 22 23
a a a a a a a a
a a a a a a a a
a a a a a a a a
-ª °« ®« °« ¬ ¯
1 ≤ i ≤ 24}
be a S-special strong 3 × 8 matrix vector space 3 × 8 natural special matrix S-field
F3×8 =1 2 8
9 10 16
17 18 24
a a ... aa a ... a
a a ... a
-ª º°« »®« »°« »¬ ¼¯
ai ∈ Q; 1 ≤
Example 4.27: Let
1 2a a
a a
-ª º°« »°« »
F4×2 =
1 2
3 4
5 6
7 8
a a
a a
a aa a
-ª º°« »°« »
®« »°« »°¬ ¼¯
ai ∈ R; 1 ≤ i ≤ 8}.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 111/344
7 8¬ ¼¯Now finally we define the S-special strong sq
vector space over the special square matrix naturafield Fn×n.
Fn×n =
11 1n
21 2n
n1 nn
a ... a
a ... a
a ... a
-ª º°« »°« »®« »°« »°¬ ¼¯
# #aij ∈ R (or Q); 1 ≤ i, j ≤
We just define this structure.
Let M =
11 12 1n
21 22 2n
n1 n2 nn
a a ... a
a a ... a
a a ... a
-ª º°« »°
« »®« »°« »°¬ ¼¯
# # # aij ∈ Q (or R), 1
be the group under addition of square matrices.
Let
11 12 1na a ... a-ª º°« »
We will illustrate this situation by some simp
Example 4.28: Let
-
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 112/344
M =1 2 3
4 5 6
7 8 9
a a a
a a a
a a a
-ª º°« »®« »°« »¬ ¼¯
ai ∈ Q; 1 ≤ i ≤
be a special strong square 3 × 3 matrix S-vectorspecial 3 × 3 square matrix natural special S-field
F3×3 =
1 2 3
4 5 6
7 8 9
a a a
a a aa a a
-ª º
°« »®« »°« »¬ ¼¯
ai ∈ Q; 1 ≤ i ≤
Example 4.29: Let
V =
1 2 3 4 5
6 7 8 9 10
21 22 23 24 25
a a a a aa a a a a
a a a a a
-ª º°« »°« »®« »°« »°¬ ¼¯
# # # # #ai ∈ R, 1
be a S-special strong 5 × 5 square matrix vectorspecial 5 × 5 natural special Smarandache field.
Example 4.30: Let
A =1 2
3 4
a a
a a
-ª º°®« »¬ ¼°̄ ai ∈ R; 1 ≤ i ≤ 4}
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 113/344
be a S-special strong 2×2 square matrix vector spaspecial 2×2 square matrix natural special S-field
F2×2 = 1 2
3 4
a a
a a
-ª º°®« »
¬ ¼°̄ai ∈ Q; 1 ≤ i ≤ 4}.
Now seen various types of S-special vector spacproceed onto define S-subspaces over natural special
DEFINITION 4.5: Let V be a S-strong special row
column matrix or m × n matrix (m≠ n) or square ma
space over the special row matrix natural S-field F
F n× n (n ≠ m) or F n× n).
Consider W ⊆ V (W a proper subset of V); if W
strong special row matrix (or column matrix or m ×
≠ n) or square matrix) S-vector space over F R (or F
F n× n) then we define W to be a S-special strong
(column matrix or m × n matrix or square matrix) v
of V over F R (or F C or F m× n or F n× n).
We will illustrate this situation by some simple ex
Example 4.32: Let
1
2
a
a-ª º°« »°« »
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 114/344
V = 2
6
a
a
°« »®« »°« »°¬ ¼¯
#ai ∈ R; 1 ≤ i ≤ 6}
be a S-special strong vector space over the S-field
FC =
1
2
6
a
a
a
-ª º°« »°« »®« »°« »°¬ ¼¯
#ai ∈ Q; 1 ≤ i ≤ 6}.
Consider
M =
1
2
3
a
0
a0
a
0
-ª º°« »°« »°« »°
« »®« »°« »°« »°« »°¬ ¼¯
ai ∈ R; 1 ≤ i ≤ 3} ⊆ V
M is a Smarandache special strong vector substhe S-field FC.
T k
P is a Smarandache special strong vector subspace ofS-field FC.
Example 4.33: Let
-ª º
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 115/344
M =1 2 3 4
5 6 7 8
9 10 11 12
a a a a
a a a a
a a a a
-ª º°« »®« »°« »¬ ¼¯
ai ∈ Q; 1 ≤ i ≤ 1
be a S-special strong vector space over the S-field
F3×4 =1 2 3 4
5 6 7 8
9 10 11 12
a a a a
a a a a
a a a a
-ª º°« »®« »°« »¬ ¼¯
ai ∈ Q; 1 ≤ i ≤ 12
P is a S-special strong vector subspace of M overF3×4.
Example 4.34: Let V = {(a1, a2, …, a9) | ai ∈ R; 1 ≤
Smarandache special strong vector space over the S-fFR = {(a1, a2, …, a9) | ai ∈ Q; 1 ≤ i ≤ 9}.
Consider M1 = {(a1, 0, a2, 0, …, 0) | a1, a2 ∈ R} ⊆S-special strong vector subspace of V over the S-field
Consider
M2 = {(0, a1, 0, a2, 0, …, 0) | ai ∈ R; 1 ≤ i ≤ 2}
It is easily verified Mi ∩ M j = (0, 0, 0, …, 0 j ≤ 4 and V = M1 + M2 + M3 + M4. Thus V is tS-strong vector subspaces.
Example 4.35: Let
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 116/344
M =
1
2
11
12
a
a
a
a
-ª º°« »°« »
°« »®« »°« »°« »°¬ ¼¯
# ai ∈ Q; 1 ≤ i ≤ 12}
be a S-special strong vector space over the S-field
FC =
1
2
11
12
a
a
a
a
-ª º°« »°« »°« »®
« »°
« »°« »°¬ ¼¯
# ai ∈ Q; 1 ≤ i ≤ 12}.
Clearly dimension of M is also 12. Consider tspecial strong vector subspaces.
1a
0
-ª º°« »°« »°
P2 =
1
0
a
0
-ª º°« »°« »
°« »°« »°®« »°« »
# ai ∈ Q; 1 ≤ i ≤ 2} ⊆ M,
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 117/344
2
0
a
0
°« »°« »°« »°« »
¬ ¼°̄
a S-strong vector subspace of M over FC.
P3 =
1
2
3
4
0
0
a
a
a
a
0
0
-ª º°« »°
« »°« »°« »°« »°« »®
« »°« »°« »
°« »°« »°« »°¬ ¼¯
#
ai ∈ Q; 1 ≤ i ≤ 4} ⊆ M,
is a S-strong special vector subspace of V over FC.
0
0
00
0
-ª º°« »°« »
°« »°« »°« »°« »
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 118/344
P4 =1
2
0
0
a
a
0
0
0
0
°« »°« »°« »®« »°
« »°« »°« »°« »°« »°« »°« »°« »
¬ ¼¯
ai ∈ Q; 1 ≤ i ≤ 2}
is again a S-strong special vector subspace of V o
P5 =
1
0
0
00
0
0
0
0
a
-ª º°« »°« »
°« »°« »°« »°« »°« »°« »®« »°
« »°« »°« »°« »°
ai ∈ Q; 1 ≤ i ≤ 2} ⊆ M
We see Pi ∩ P j =
0
0
0
0
ª º« »« »
« »« »« »« »¬ ¼
# if i ≠ j; 1 ≤ i, j ≤ 5 and
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 119/344
0« »¬ ¼V = P1 + P2 + P3 + P4 + P5.
Thus V is a direct sum of S-special strong vector Example 4.36: Let
M =
1 2 3
4 5 6
7 8 9
10 11 12
a a a
a a a
a a a
a a a
-ª º°« »°
« »®« »°« »°¬ ¼¯
ai ∈ R; 1 ≤ i ≤ 12}
be a S-strong special vector space over the S-field,
F4×3 =
1 2 3
4 5 6
7 8 9
10 11 12
a a a
a a a
a a a
a a a
-ª º°« »°« »®« »°« »°¬ ¼¯
ai ∈ R; 1 ≤ i ≤ 12
1 2a a 0
0 0 0
-ª º°« »°
B2 =
1 2
3 4
a 0 a
0 a a
0 0 00 0 0
-ª º°« »°« »®
« »°« »°¬ ¼¯
ai ∈ R; 1 ≤ i ≤ 4
b S i l b f M F
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 120/344
be a S-strong special vector subspace of M over F
B3 =
1
2
3
a 0 0
a 0 0a 0 0
0 0 0
-ª º
°« »°« »®« »°« »°¬ ¼¯
ai ∈ R; 1 ≤ i ≤ 3}
is a S-strong special vector subspace of M over F
Finally
B4 =
1
2 3
4
a 0 0
0 0 0
0 a a
a 0 0
-ª º°« »°« »®« »°« »°
¬ ¼¯
ai ∈ R; 1 ≤ i ≤ 4}
is again a S-strong special vector subspace of V o
We see Bi ∩ B j ≠
0 0 0
0 0 0
0 0 00 0 0
ª º« »« »
« »« »¬ ¼
; even if i ≠ j, 1
they may find applications in all places where the rereal number (or a rational number) but an array of nu
We can define orthogonal vectors of S-special stvector spaces also.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 121/344
First we see how orthogonal vector matrices when they are defined over R or Q or C.
Let VR = {(a1, …, an) | ai ∈ Q; 1 ≤ i ≤ n} be avector space defined over the field Q.
We define for any x = (a1, …, an) and y = (b1, …x is perpendicular to y if x ×n y = (0).
Thus if VR = {(x1, x2, x3, x4, x5) | xi ∈ R; 1 ≤ i ≤matrix vector space defined over Q and if x = (0, 4, -y = (1, 0, 0, 8, 0) are in VR. We see x ×n y = orthogonal with y.
VC =1
n
a
a
-ª º°« »®« »
°« »¬ ¼¯
# ai ∈ Q, (or R); 1 ≤ i ≤ n} be the v
of column matrices over Q (or R) respectively.
1
2
x
x
ª º« »« »
1
2
y
y
ª º« «
For instance if x =
2
1
00
0
ª º« »−« »
« »« »« »« »« »
and y =
0
0
12
3
ª º« »« »
« »« »« »« »« »
then
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 122/344
0
7« »« »¬ ¼
3
0« »« »¬ ¼
x ×n y =
21
0
0
0
7
ª º« »−« »« »« »« »« »
« »« »¬ ¼
×n
00
1
2
3
0
ª º« »« »« »« »« »« »
« »« »¬ ¼
=
00
0
0
0
0
ª º« »« »« »« »« »« »
« »« »¬ ¼
.
Now we can define, unlike in other matrix case of these vector spaces Vm×n (m ≠ n)orthogonality under natural product. This is a
enjoyed only by vector spaces on which naturaldefined.
We just illustrate this situation by some exam
Example 4.37: Let
1 2 3a a a
a a a
-ª º°« »°
Now let
x =
3 2 0
0 1 5
1 1 0
ª º
« »« »« »« »
and y =
0 0 7
9 0 0
0 0 9
ª º
« »« »« »« »
be in V
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 123/344
2 0 7
0 1 8
« »« »« »¬ ¼
0 8 0
4 0 0
« »« »« »¬ ¼
we see x ×n y =
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
ª º« »« »« »« »« »
« »¬ ¼
thus we say x is orth
under natural product in V5×3.
It is pertinent to mention here that we can havein V5×3 such that for a given x in V5×3. x ×n y = (0).
Now we see all elements in V5×3 are orthog
natural product to the zero 5 × 3 matrix
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
ª º« »« »« »« »
« »« »¬ ¼
.
Example 4.38: Let
Take x = 1 3 5
2 4 6
a a a 0 0 0
a a a 0 0 0
ª º« »¬ ¼
and
y = 1 3
2 4
0 0 0 a 0 a
0 0 0 a 0 a
ª º« »¬ ¼
in V2×6. ai ∈
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 124/344
¬ ¼
We see x ×n y =0 0 0 0 0 0
0 0 0 0 0 0
ª « ¬
Thus x is orthogonal with y. Infact we hay’s which are orthogonal with x.
Example 4.39: Let
V = 1 2
3 4
a a
a a
-ª º°®« »
¬ ¼°̄ai ∈ R; 1 ≤ i ≤ 4}
be a 2 × 2 matrix vector space over the field R. ×product on V.
We define two matrices in V to be orthogonal
x ×n y =0 0
0 0
ª º« »¬ ¼
for y, x ∈ V. We see x =
then y = 10 b
0 0
ª º« »¬ ¼
is orthogonal with x.
Further 1
2
0 b
b 0
ª º« »¬ ¼
= b, is orthogonal with x un
product as x ×n b = 0 00 0
ª º« »¬ ¼
.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 125/344
Now x⊥ = 1 1
2 1
0 b 0 00 0 0 b, , ,
b 0 b 00 0 0 0
- § · § ·§ · § ·° ® ¨ ¸ ¨ ¨ ¸ ¨ ¸° © ¹ © ¹© ¹ © ¹¯
x⊥ is additively closed and also ×n product also is clx⊥ is a proper subspace of V defined as perpendicular with x.
Consider x =0 a
b 0
ª º« »¬ ¼
in V; now the elements pe
with x are0 0 t 0 0 0 t 0
, , ,0 0 0 0 0 u 0 u
- ½§ · § · § · § ·° °® ¾¨ ¸ ̈ ¸ ̈ ¸ ̈ ¸° °© ¹ © ¹ © ¹ © ¹¯ ¿
.
We see this is also a subspace of V.
Infact if
B =0 0 t 0 0 0 x 0
, , ,0 0 0 0 0 u 0 y
- ½ª º ª º ª º ª º° °® ¾« » « » « » « »° °¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼¯ ¿
⊆
and
Example 4.40: Let
M =1 2 3
4 5 6
7 8 9
a a a
a a a
a a a
-ª º
°« »®« »°« »¬ ¼¯
ai ∈ Q; 1 ≤ i ≤
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 126/344
be a 3 × 3 vector space over the field Q. Conside
x =
a b c
0 0 0
0 0 d
ª º« »« »« »¬ ¼
in M.
The elements perpendicular to x be denoted byx⊥ = B =
1 2
0 0 0 0 0 0 0 0 0
0 0 0 , 0 0 a , a 0 0
0 0 0 0 0 0 0 0 0
-ª º ª º ª º°« » « » « »®« » « » « »°« » « » « »¬ ¼ ¬ ¼ ¬ ¼¯
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 , 0 0 0 , a b 0 , a 0 b
d 0 0 0 e 0 0 0 0 0 0 0
ª º ª º ª º ª º« » « » « » « « » « » « » « « » « » « » « ¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼
0 0 0 0 0 0 0 0 0 0 0 00 0 a , 0 0 a , a 0 0 , a 0 0
b 0 0 0 b 0 b 0 0 0 b 0
ª º ª º ª º ª º« » « » « » « « » « » « » « « » « » « » «¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼
0 0 0 0 0 0 0 0 0 0 0 0 0
0 a b , 0 a b , a 0 b , a 0 b , a
c 0 0 0 c 0 c 0 0 0 c 0 b
ª º ª º ª º ª º ª« » « » « » « » «« » « » « » « » «« » « » « » « » «¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼ ¬
0 0 0 0 0 0 0 0 0 0 0 0 0ª º ª º ª º ª º ª« » « » « » « » «
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 127/344
0 a 0 , 0 0 a , a b c , a b c , a
b c 0 b c 0 d b 0 d 0 0 0
« » « » « » « » «« » « » « » « » «« » « » « » « » «¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼ ¬
0 0 0 0 0 0 0 0 0
0 a b , a 0 b , a b 0
c d 0 d c 0 d c 0
½ª º ª º ª º°« » « » « »¾« » « » « »°« » « » « »¬ ¼ ¬ ¼ ¬ ¼¿
is again a subspace orthogonal with x.
Inview of this we have the following theorem.
THEOREM 4.1: Let V be a matrix vector space ove
Let 0 ≠ x ∈ V. The elements orthogonal to x un
product × n is a subspace of V over F.
Proof: Let 0 ≠ x ∈ V. Suppose x⊥ = B = {y ∈ V | yto show B is a subspace of V.
Clearly B ⊆ V, by the very definition of orthogoof x. Further (0) ∈ V is such that x ×n (0) = (0orthogonal with x. Now let z, y ∈ B be orthogonal w
so y + z ∈ B. Also if y ∈ B is such that x ×n y = = 0 so if y ∈ B then –y ∈ B. Finally let a ∈
show ay ∈ B. Consider x ×n ay = a (x ×n y) = a.0
Thus B ⊆ V is a vector subspace of V.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 128/344
Now we can define orthogonality of two elevector spaces under natural product, we now
properties associated with orthogonal natural prod
Example 4.41: Let
W =
a b c
d e f g h i
-ª º°« »
®« »°« »¬ ¼¯a, b, c, d, e, f, g, h, i
be a vector space over Q.
Consider x⊥ = B =
0 0 0 0 0 0 0 0 0 0
0 0 0 , 0 0 0 , 0 0 0 , 0
0 0 0 a 0 0 0 b 0 0
-ª º ª º ª º ª °« » « » « » « ®« » « » « » « °« » « » « » «
¬ ¼ ¬ ¼ ¬ ¼ ¬ ¯
0 0 0 0 0 0 0 0 0 0 0ª º ª º ª º ª
B⊥=
0 0 0 a 0 0 0 b 0 0 0
0 0 0 , 0 0 0 , 0 0 0 , 0 0
0 0 0 0 0 0 0 0 0 0 0
-ª º ª º ª º ª °« » « » « » « ®« » « » « » « °
« » « » « » « ¬ ¼ ¬ ¼ ¬ ¼ ¬ ¯
0 0 0 0 0 0 0 0 0 0 0 0 aª º ª º ª º ª º ª« » « » « » « » «
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 129/344
0 0 0 , d 0 0 , 0 e 0 , 0 0 f , 0
0 0 0 0 0 0 0 0 0 0 0 0 0
« » « » « » « » «« » « » « » « » «« » « » « » « » «¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼ ¬
0 a b a 0 b a 0 0 0 0 0 0
0 0 0 , 0 0 0 , b 0 0 , a b 0 , a
0 0 0 0 0 0 0 0 0 0 0 0 0
ª º ª º ª º ª º ª« » « » « » « » «« » « » « » « » «« » « » « » « » «¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼ ¬
0 0 0 a 0 0 a 0 0 0 a 0 00 a b , 0 b 0 , 0 0 b , b 0 0 , 0
0 0 0 0 0 0 0 0 0 0 0 0 0
ª º ª º ª º ª º ª« » « » « » « » «« » « » « » « » «« » « » « » « » «¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼ ¬
0 a 0 0 0 a 0 0 a 0 0 a a
0 0 b , b 0 0 , 0 b 0 , 0 0 b , 0
0 0 0 0 0 0 0 0 0 0 0 0 0
ª º ª º ª º ª º ª« » « » « » « » «« » « » « » « » «« » « » « » « » «¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼ ¬
0 0 0 a 0 0 a 0 0 a 0 b a
a b c , 0 b c , b c 0 , 0 c 0 , b0 0 0 0 0 0 0 0 0 0 0 0 0
ª º ª º ª º ª º ª« » « » « » « » «
« » « » « » « » «« » « » « » « » «¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼ ¬
0 0 a 0 a b 0 a b 0 a b
0 b c , c 0 0 , 0 c 0 , 0 0 c
0 0 0 0 0 0 0 0 0 0 0 0
ª º ª º ª º ª º« » « » « » «
« » « » « » « « » « » « » « ¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼
a b c a b c 0 a b 0 a bª º ª º ª º ª º
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 130/344
a b c a b c 0 a b 0 a b
0 d 0 , 0 0 d , d 0 c , d c 0
0 0 0 0 0 0 0 0 0 0 0 0
ª º ª º ª º ª º« » « » « » « « » « » « » « « » « » « » « ¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼
a 0 0 0 a 0 0 0 a a b 0
b c d , b c d , b c d , c d 0
0 0 0 0 0 0 0 0 0 0 0 0
ª º ª º ª º ª º« » « » « » « « » « » « » « « » « » « » « ¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼
a 0 b a 0 b a b 0 a b c
0 c d , c d 0 , c 0 d , 0 d e
0 0 0 0 0 0 0 0 0 0 0 0
ª º ª º ª º ª º« » « » « » « « » « » « » « « » « » « » « ¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼
0 a b a b 0 a 0 b ac d e , c d e , d e c , d
0 0 0 0 0 0 0 0 0 0
ª º ª º ª º ª « » « » « » « « » « » « » « « » « » « » « ¬ ¼ ¬ ¼ ¬ ¼ ¬
We see B⊥ ⊕ B = W.
Thus we have the following theorem.
COROLLARY 4.1: Let V be a matrix vector space ov
F. {0} ∈ V; the space perpendicular to V under natu
× n is V, that is {0}⊥
= V.
COROLLARY 4.2: Let V be a matrix vector space ov
F. The space perpendicular to V under natural pro
that is {V}⊥
= {0}
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 131/344
that is {V} {0}.
Example 4.42: Let
V = 1 2 3
4 5 6
a a a
a a a
-ª º°®« »
¬ ¼°̄ai ∈ Q; 1 ≤ i ≤ 6}
be a vector space over Q.
Now consider x = 1 2
3 4
0 a a
0 a a
ª º« »¬ ¼
be the element
vectors perpendicular or orthogonal to x are given by
a 0 0 0 0 0 a 0 0 0 0 0, , ,0 0 0 a 0 0 b 0 0 0 0 0
- ª º ª º ª º ª º° ® « » « » « » « »° ¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼¯
Now take y =a 0 0
b 0 0
ª º« »¬ ¼
∈ B, the vectors perp
y are y⊥ =
-
0 a 0 0 0 c 0 a b 0 a b 0, , , ,
0 0 b 0 d 0 0 c 0 0 0 b 0
ª º ª º ª º ª º ª« » « » « » « » «¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼ ¬
We see x ∈ y⊥ under natural product. Nowx ×n y = (0)} and ¢y⊥² = {x ∈ V | x ×n y = {0subspaces of V but are such that ¢x⊥² ∪ ¢y⊥² = V
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 132/344
p ¢ ² ¢y ²= {(0)}.
Further x = 0 a b0 c d
ª º« »¬ ¼
is in ¢y
Suppose z⊥ =a 0 0
0 0 0
ª º« »¬ ¼
∈ B is tak
z⊥ = {m ∈ V | m ×n z = (0)}.
=0 0 0 0 a 0 0 0 b 0 0 0
, , ,0 0 0 0 0 0 0 0 0 x 0 0
-ª º ª º ª º ª °®« » « » « » « °¬ ¼ ¬ ¼ ¬ ¼ ¬ ¯
0 0 0 0 a b 0 b 0 0 a 0, , ,
0 0 t 0 0 0 a 0 0 0 b 0
ª º ª º ª º ª º« » « » « » « ¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼
0 0 a 0 0 a 0 0 a 0 0 0
, , ,b 0 0 0 b 0 0 0 b a b 0
ª º ª º ª º ª º
« » « » « » « ¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼
0 0 0 0 a b 0 a b 0 a b 0 0, , , ,
a b c 0 c d c 0 d c d 0 b c
ª º ª º ª º ª º ª « » « » « » « » « ¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼ ¬
0 a 0 0 a b,
b c d d c x
½ª º ª º°¾« » « »°¬ ¼ ¬ ¼¿
= T.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 133/344
We see though z ∈ B still ¢z⊥² ≠ ¢y⊥².
Further every element in T is not perpendicularthe natural product ×n.
For consider m =0 a 0
b c d
ª º« »¬ ¼
in T and
x ×n m =0 a b
0 c d
ª º« »¬ ¼
×n 0 a 0
b c d
ª º« »¬ ¼
=
1
1 1
0 x 0
0 y z
ª º
« »¬ ¼ ≠
0 0 0
0 0 0
ª º
« »¬ ¼ .
Thus we can say for any x ∈ V we have one andin V such that x is the complement of y with respecproduct ×n.
We say complement, if x⊥ generates the spacegenerates another space say C.
THEOREM 4.3: Let V be a matrix vector space
(or R). Let × n be the natural product defined on
y is the main complement of V and vice versa, th
V and ¢ x⊥
² ∩ ¢ y⊥
² = (0).
(1) However for no other element t in ¢
main complement of x.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 134/344
p f
(2) Also no element in ¢ x⊥ ² will
complement of y only x will complement of y.
The proof is left as an exercise. However wsituation by some example.
Example 4.43: Let
M =a b
c d
-ª º°®« »
¬ ¼°̄a, b, c, d ∈ Q}
be the vector space of 4 × 4 matrices over the fiel
Take p =x 0
y 0
ª º« »¬ ¼
∈ M, now the complem
natural product in M are
0 0 0 a 0 a
, ,0 0 0 b 0 0
-ª º ª º ª °
®« » « » « °¬ ¼ ¬ ¼ ¬ ¯
0 0ª º
Now the complements of q under natural pro
M are0 0 a 0 a 0 0 0
, , ,0 0 b 0 0 0 a 0
-ª º ª º ª º ª º°®« » « » « » « »
¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼°̄
a, b ∈ Q}.
We see V + T = M and V ∩ T =0 0
0 0
ª º« »¬ ¼
.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 135/344
0 0¬ ¼
Now x = 0 a0 0ª º« »
¬ ¼is in T. We find the elements
to x in M under the natural product ×n.
¢x⊥² =0 a
0 0
⊥
ª º« »¬ ¼
=
0 0 a 0 0 0 0 0, , , ,
0 0 0 0 b 0 0 c
-ª º ª º ª º ª º°®« » « » « » « »°¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼¯
a 0 a 0 0 0 a 0, , ,b 0 0 c b c b c
½ª º ª º ª º ª º°¾« » « » « » « »°¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼¿ .
The main complement of x in M isa 0
b c
ª º« »¬ ¼
oth
complements.Consider
Example 4.44: Let
V =
1
2
8
a
a
a
-ª º
°« »°« »®« »°« »°¬ ¼¯
#ai ∈ Q; 1 ≤ i ≤ 8}
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 136/344
¬ ¼¯
be a vector space of column matrices over the fiel
Consider the element a =
1
2
3
a
a
0
00
0
0
a
ª º« »« »« »« »
« »« »« »« »« »« »« »¬ ¼
in V.
To find complements or elements orthogonal
¢a²⊥ =
1
2
0 0 0 00
0 0 0 0 00
0 0 0 0 0 a00 a 0 0 0 a
, , , , , ,a
ª º ª º ª º ªª º « » « » « » «ª º « » « » « » « » «« »« » « » « » « » «ª º « » « » « » « » « » «« » « » « » « » « » « » «« » « » « » « » « » « » «« »#
1
0
0
a0
,
- º ª ° » « ° » «
° » « ° » « ° » « ® » «
1 1
1 1 1
1 2
2 2 2
0 0 0 0 0 0
0 0 0 0 0 0
a a 0 0 0 0
0 0 0 a a a, , , , ,
0 0 a a 0 0
a 0 a 0 a 0
ª º ª º ª º ª º ª º ª« » « » « » « » « » «« » « » « » « » « » «« » « » « » « » « » «« » « » « » « » « »« » « » « » « » « »« » « » « » « » « »« » « » « » « » « »« » « » « » « » « »
1
1
0 0
0 0
0 0
0 0, ,
a 0
0 a
º ª º ª » « » « » « » « » « » «
« » « » « « » « » « « » « » « « » « » « « » « » «
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 137/344
2 2 2
2 2
a 0 a 0 a 0
0 a 0 0 0 a
0 0 0 0 0 0
« » « » « » « » « »« » « » « » « » « »« » « » « » « » « »
« » « » « » « » « »¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼ ¬
1
2 2
0 a
a a
0 0
« » « » « « » « » « « » « » «
« » « » « ¼ ¬ ¼ ¬
1 1 1 1 1 1
2 2 2
3 2 2
3 3 2
3 3 3
0 0 0 0 0 0
0 0 0 0 0 0
a a a a a a
a a a 0 0 0, , , , ,a 0 0 a 0 a
0 a 0 a a 0
0 0 a 0 a a
0 0 0 0 0
ª º ª º ª º ª º ª º« » « » « » « » « »« » « » « » « » « »« » « » « » « » « »
« » « » « » « » « »« » « » « » « » « »« » « » « » « » « »« » « » « » « » « »« » « » « » « » « »« » « » « » « » « »« » « » « » « » « »« » « » « » « » « »¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼
1 1
2
3 2
3
0 0
0 0
0 0
a a, ,a 0
a a
0 a
0 0 0
ª º ª º ª « » « » « « » « » « « » « » «
« » « » « « » « » « « » « » « « » « » « « » « » « « » « » « « » « » « « » « » « ¬ ¼ ¬ ¼ ¬
1 1 1
1 2 2 1
2 1 3 3 2 2
0 0 0 0 0 0
0 0 0 0 0 0
0 0 a a a 0
a 0 a a 0 a, , , , ,a a a a a a
ª º ª º ª º ª º ª º« » « » « » « » « »« » « » « » « » « »« » « » « » « » « »« » « » « » « » « »« » « » « » « » « »« » « » « » « » « »« » « » « » « » « »
1
2
3
0
0
a
a,a
½ª º ª º°« » « »°« » « »°« » « »°« » « »°« » « »¾« » « »°« » « »°
.
The main complement of
1
2
a
a
0
0
0
0
ª º« »« »« »« »« »« »« »« »
is
1
2
3
4
0
0
a
a
a
a
ª º« »« »« »« »« »« »« »« »
.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 138/344
3
0
a
« »« »« »
« »¬ ¼
4
5
a
a
0
« »« »« »
« »¬ ¼
Certainly this type of study will be a bocoding theory as in case of algebraic coding thuse only matrices which are m × n (m ≠ n) matrix and generator matrix.
Now we define other related properties ofwith natural product on them. Suppose S is a suvector space defined on R or Q we can define
S⊥ = {x ∈ V | x ×n s = (0) for every
We will illustrate this situation by some simp
Example 4.45: Let
V =
1 2
3 4
a aa a
-ª º°« »°« »®« »
ai ∈ Q; 1 ≤ i ≤ 8
Consider
S =
a b 0 0
0 0 0 0,0 0 0 0
0 0 c d
-ª º ª º
°« » « »°« » « »®« » « »°« » « »°¬ ¼ ¬ ¼¯
a, b, c, d ∈ Q} ⊆ V
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 139/344
To find S⊥ . S⊥ =
0 0 0 0 0 0 0 0
0 0 a b a 0 0 a, , ,
0 0 d e 0 0 0 0
0 0 0 0 0 0 0 0
-ª º ª º ª º ª °« » « » « » « °« » « » « » « ®« » « » « » « °« » « » « » « °¬ ¼ ¬ ¼ ¬ ¼ ¬ ¯
0 0 0 0 0 0 0 00 0 0 0 0 0 a b
, , , ,b 0 0 d a b 0 0
0 0 0 0 0 0 0 0
ª º ª º ª º ª º« » « » « » « »« » « » « » « »« » « » « » « »« » « » « » « »¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼
0 0 0 0 0 0 0 0a 0 a 0 0 a 0 b
, , , ,0 b b 0 0 b a 0
0 0 0 0 0 0 0 0
ª º ª º ª º ª º« » « » « » « »« » « » « » « »« » « » « » « »« » « » « » « »¬ ¼ ¬ ¼ ¬ ¼ ¬ ¼
0 0 0 0 0 0 0 0
a b a b 0 a a 0
½ª º ª º ª º ª º°« » « » « » « »°« » « » « » « »
We see S⊥ is a subspace of V. Fu
complement of x =
a b
0 0
0 0
0 0
ª º« »
« »« »« »¬ ¼
and
0 0
0 0
0 0
b c
ª º« »
« »« »« »¬ ¼
= y are
0 0ª º« »
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 140/344
main complement of x isa b
c de f
« »« »
« »« »¬ ¼
and that of y
the main complement of x is not orthogonal with
=
0 0
0 00 0
b c
ª º« »« »« »« »¬ ¼
0 0
a bc d
e f
ª º« »« »« »« »¬ ¼
=
0 0
0 00 0
be cf
ª º« »« »« »« »¬ ¼
≠
0
00
0
ª « « « « ¬
Similarly the main complement of
0 00 0
0 0
a b
ª º« »« »« »« »¬ ¼
,
is not orthogonal with
a b
0 0
0 0
ª º
« »« »« »« »
under natural prod
We can define as in case of usual vector spaces d
transformation for matrix vector spaces. However it
less to define linear transformation in case of S-spvector spaces defined over the S-field. However ionly linear operators can be defined. The definitionetc in case of the former vector space is a matter of we see no difference with usual spaces. However
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 141/344
we see no difference with usual spaces. Howeverlatter S-special strong vector spaces over a S-field we
only linear operators.We just illustrate this situation by an example.
Example 4.46: Let
V =
1 2
3 4
5 6
7 8
a aa a
a a
a a
-ª º°« »°« »®« »°« »°¬ ¼¯
ai ∈ R; 1 ≤ i ≤ 8}
be a S-special super vector space over the S-field.
F4×2 =
a b
c d
e f
g h
-ª º°« »°« »®« »
°« »°¬ ¼¯
a, b, c, d, e, f, g, h ∈ Q}
It is easily verified η is a linear operator on V.
We can find kernel η =
a b a
c d cVe f e
g h g
- ª º §
° ¨ « »° ¨ « » ∈ η ® ¨ « »° ¨ « »° ¬ ¼ ©¯
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 142/344
=
0 a
b 00 c
d 0
-ª º°« »°« »®« »°« »°¬ ¼¯
a, b, c, d ∈
is a subspace of V. Now interested reader can woperators on S-special strong vector spaces over t
Thus all matrix vector spaces are linear algnatural product. Now as in case of usual vectofor the case of matrix vector spaces also definlinear functional. But in case of S-strong specispaces we can not define only Smarandache line
matter of routine as it needs more modifications a
Now we have discussed some of propertiespaces. We now define n - row matrix vector spF.
DEFINITION 4.6: Let
V = {(a a ) | a = (x x ); x
Example 4.47: Let V = {(a1, a2, a3, a4) | a j = (x1, xwhere xi ∈ Q, 1 ≤ i ≤ 5 and 1 ≤ j ≤ 4} be a 5-structured vector space over Q.
We will just show how addition and scalar multperformed on V.
Suppose x = ((3, 0, 2, 4, 5), (0, 0, 0, 1, 2), (1, 1,
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 143/344
pp (( , , , , ), ( , , , , ), ( , ,0, 1, 0, 5)) ∈ V and a = 7 then 7x = ((21, 0, 14, 28, 3
7, 14), (7, 7, 7, 21, 0), (14, 0, 7, 0, 35)) ∈ V.
Let y = ((4, 0, 1, 1, 1), (0, 1, 0, 1, 2), (1, 0, 1, 1, 0, 1)) ∈ V then x + y = ((7, 0, 3, 5, 6), (0, 1, 0, 2, 4)1), (4, 0, 1, 0, 6)) ∈ V. We see V is a row matrixvector space over Q.
Example 4.48: Let
P = {(a1, a2, a3) | ai = (x1, x2, …, x15) x j ∈ Q1 ≤ i ≤ 3; 1 ≤ j ≤ 15}
a row matrix structured vector space over Q. Werow matrix structured subvector space as in case of uspace. On P we can always define the natural prodis always a row matrix structured linear algebranatural product ×n.
Example 4.49: Let
V {( ) | ( ) Q 1 ≤ j ≤ 3
is a row matrix structured vector subspace of Vreader can see the difference between the subspac
Let V = {(x1, …, xn) | xi ∈ R+
∪ {0}, 1semigroup under addition. V is a semivector semifield R+ ∪ {0} or Q+ ∪ {0} or Z+ ∪ {0}.
x-ª º
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 144/344
Likewise M =
1
2
m
x
x
x
-ª º°« »°
« »®« »°« »°¬ ¼¯
# xi ∈ Q+ ∪ {0}, 1
semigroup under addition. M is a semivector semifield Z+ ∪ {0} or Q+ ∪ {0}. M is not a s
over R
+
∪ {0}.
P =
11 12 1n
21 22 2n
m1 m 2 mn
a a ... a
a a ... a
a a ... a
-ª º°« »°« »®« »°« »
°¬ ¼¯
# # #aij ∈ Z+ ∪
1 ≤ i ≤ m, 1 ≤ j ≤ n}
is a semivector space over the semifield Z+ ∪ {0}
T =
11 12 1n
21 22 2n
a a ... aa a ... a
-ª º°« »°« »®« »# # #
aij ∈ Q+ ∪ {0},
Example 4.50: Let
V = {(x1, x2, x3, x4, x5, x6) | xi ∈ 3Z
+
∪ {0}; 1 ≤
be the semivector space over the semifield S = Z+ ∪ {
V is also a semilinear algebra over the semifield S
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 145/344
g
Example 4.51: LetV1 = {(x1, x2, x3, x4, x5, x6) where xi ∈ R+ ∪ {0}; 1
be the semivector space over the semifield S = Z+ ∪ {
It is interesting to compare V and V1 for V
dimensional where as V2 is of infinite dimension.
Example 4.52: Let
V =
1
2
3
4
5
6
7
8
x
xx
x
x
x
xx
-ª º°« »
°« »°« »°« »°« »®« »°« »°« »°
« »°« »°« »¬ ¼¯
xi ∈ Z+ ∪ {0}, 1 ≤ i ≤ 8}
Example 4.53: Let
M =
1
2
3
4
x
xx
x
x
-ª º
°« »°« »°« »°« »°« »®« »°« »
xi ∈ Q+ ∪ {0}, 1 ≤ i ≤
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 146/344
5
6
7
8
x
x
x
x
« »°« »°
« »°« »°« »°« »¬ ¼¯
be a semivector space over the semifield S = Z+
dimension of M over S is infinite. Example 4.54: Let
M =
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
17 18 19 20
21 22 23 24
25 26 27 28
29 30 31 32
a a a a
a a a a
a a a a
a a a a
a a a a
a a a a
a a a aa a a a
-ª º°« »°
« »°« »°« »°« »°« »°« »®
« »°« »°« »°« »°« »°
ai ∈ Z+ ∪ {0},
Example 4.55: Let
M =
1 2 3
4 5 6
7 8 9
a a a
a a aa a a
-ª º
°« »®« »°« »¬ ¼¯
ai ∈ Q+ ∪ {0}, 1 ≤ i ≤
be a semivector space over the semifield S = Z+ ∪ {0
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 147/344
p {
Take
P =1 2 3
4 5
6
a a a
a 0 a
a 0 0
-ª º°« »®« »°« »¬ ¼¯
ai ∈ Q+ ∪ {0}, 1 ≤ i ≤
P is a semivector subspace of V over S = Z+ ∪ {0}.
Example 4.56 : Let
M =
1 2 3
4 5 6
7 8 9
10 11 12
13 14 15
16 17 18
a a a
a a aa a a
a a a
a a a
a a a
-ª º°
« »°« »°« »°
« »®« »°« »°« »°« »°¬ ¼
¯
ai ∈ Q+ ∪ {0}, 1 ≤ i ≤
be a semivector space over the semifield S = Q+ ∪ {0
M2 =
1 2 3
4 5 6
0 0 0
a a a
a a a
0 0 0
0 0 0
0 0 0
-ª º°« »°« »°« »°« »®
« »°« »°« »°« »°¬ ¼¯
ai ∈ Q+ ∪ {0}, 1 ≤ i
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 148/344
is a semivector subspace of M over S = Q+ ∪ {0}
M3 =1 2 3
4 5 6
0 0 0
0 0 0
0 0 0
a a a
a a a0 0 0
-ª º°« »°« »°« »°
« »®« »°« »°« »°« »°¬ ¼¯
ai ∈ Q+ ∪ {0}, 1 ≤ i
is a semivector subspace of M over S = Q+ ∪ {0}
M4 =
1 2 3
4 5 6
0 0 0
0 0 0
0 0 0
0 0 0
a a a
a a a
-ª º°« »°« »°« »°
« »®« »°« »°« »°« »°¬ ¼¯
ai ∈ Q+ ∪ {0}, 1 ≤ i
is the semivector subspace of M over S.
Thus M is the direct sum of semivector subspover S.
Example 4.57:Let
V =1 2 3 4 5
6 7 8 9 10
a a a a a
a a a a a
a a a a a
-ª º°« »®« »°« »¬ ¼¯
ai ∈ Z+ ∪ {0}, 1
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 149/344
11 12 13 14 15a a a a a« »¬ ¼¯
be a semivector space over the semifield S = Z+
∪ {0
Consider
P1 =
1
2
3
a 0 0 0 0
a 0 0 0 0a 0 0 0 0
-ª º°« »
®« »°« »¬ ¼¯ai ∈ Z
+
∪ {0}, 1 ≤ i ≤
be a semivector subspace of V over S.
Let
P2 =4 1 2
3
a a a 0 0
0 0 a 0 0
0 0 0 0 0
-ª º°« »®« »°« »¬ ¼¯
ai ∈ Z+ ∪ {0}, 1 ≤ i ≤
be a semivector subspace of V over S.
Consider
Further
P4 =
1 2
3 4
a 0 a 0 0
0 0 0 0 0a 0 a 0 0
-ª º°
« »®« »°« »¬ ¼¯
ai ∈ Z+ ∪ {0}, 1 ≤
is a semivector subspace of V over S.
-
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 150/344
P5 =1 3 4
2
5
a 0 0 a a
0 0 0 a 0
0 a 0 0 0
-ª º
°« »®« »°« »¬ ¼¯
ai ∈ Z+ ∪ {0}, 1
is a semivector subspace of V over S.
P6 =1
5
2 3 4
a 0 0 0 0
0 0 0 0 a
0 0 a a a
-ª º°« »®« »
°« »¬ ¼¯
ai ∈ Z+ ∪ {0}, 1
is a semivector subspace of V over S.
We see Pi ∩ P j ≠ 0 0 0 0 00 0 0 0 0
0 0 0 0 0
ª º« »« »« »¬ ¼
if i ≠ j,
We see V ⊆ P1 + P2 + P3 + P4 + P5 + P6; thudirect sum of semivector subspaces.
Now we have seen examples of direct sudirect sum of semivector subspaces over the sem
Example 4.58: Let
V = ii
i 0
a x∞
=
-®¯¦
ai = (x1, x2, x3, x4) | x j ∈ Z+ ∪ {0}; 1
be a semivector space of infinite dimension over S = Z
Clearly V is also a semilinear algebra over Snatural product ×
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 151/344
natural product ×n.
Example 4.59: Let
M = ii
i 0
a x∞
=
-®¯¦ ai =
1
2
3
10
x
x
x
x
ª º« »« »« »
« »« »« »¬ ¼
#
where x j ∈ Z+ ∪ {0}; 1
be a semivector space of infinite dimension over S = Z
Example 4.60: Let
P = ii
i 0
d x∞
=
-®¯¦ di =
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
a a a a
a a a a
a a a a
a a a a
ª º« »« »« »« »¬ ¼
where ai ∈ Q+ ∪ {0}; 1 ≤ i ≤ 16}
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 152/344
Example 4.64: Let
V =
1
2
3
4
aa
a
a
-ª º°« »°« »°« »®
« »°« »°« »°¬ ¼¯
with ai ∈ Z+ ∪ {0}, 1 ≤ i ≤ 5}
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 153/344
5a« »°¬ ¼¯
be a semivector space over the semifield S = Z+ ∪ {0
Consider
M1 = 1
2
00
a
a
0
-ª º°« »°« »°« »®
« »°« »°« »°¬ ¼¯
with ai ∈ Z+ ∪ {0}, 1 ≤ i ≤ 2} ⊆
be a semivector subspace of V over S.
M2 =
1
2
3
a
a
0
0a
-ª º°« »°« »°« »®
« »°« »°« »°¬ ¼¯
ai ∈ Z+ ∪ {0}, 1 ≤ i ≤ 3} ⊆ V,
Consider P1 = {(0, 0, 0, a1, 0, 0, 0, 0, 0, a7) 1 ≤ i ≤ 7} ⊆ V be a semivector subspace of V ove
P2 = {(a1, a2, 0,0, …, 0) | a1, a2 ∈ Z
+
∪ {0}be a semivector subspace of V over S.
P3 = {(0, 0, a1, 0, a2, a3, 0,0,0,0) | ai ∈ Z+ ∪ {0},be a semivector subspace of V over S.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 154/344
P4 = {(0, 0, 0, 0, 0, 0, a1, a2, a3, 0) | ai ∈ Z+
∪ {0}is again a semivector subspace of V over S.
We see every vector in Pi is orthogonal wvector in P if i ≠ j; 1 ≤ i, j ≤ 4.
Further V = P1 + P2 + P3 + P4 and Pi ∩ P j = (
Example 4.66: Let
M =
1 2 3
4 5 6
7 8 9
10 11 12
13 14 15
a a a
a a aa a a
a a a
a a a
-ª º°« »
°« »°« »®« »°« »°« »°¬ ¼¯
ai ∈ Z+ ∪ {0}, 1 ≤
be a semivector space over the semifield S = Z+ ∪
Take
P2 =
1 2 3
4 5 6
7 8 9
a a a
0 0 0
a a a
0 0 0
a a a
-ª º°« »°« »°« »®
« »°« »°« »°¬ ¼¯
ai ∈ Z+ ∪ {0}, 1 ≤ i ≤ 9}
is a semivector subspace of M over S = Z+ ∪ {0}.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 155/344
We see for every x ∈ P1 we have x ×n y = (0) foP2.
Thus M = M1 + M2 and P1 ∩ P2 = (0). We say tis orthogonal with the space P2 of M.
However if
P3 =
1 2 3
0 0 0
0 0 0
0 0 0
0 0 0a a a
-ª º°« »°« »°« »®
« »°« »°« »°¬ ¼¯
ai ∈ Z+ ∪ {0}, 1 ≤ i ≤ 3}
we see P1 and P3 are such that for every x ∈ P1 we ha(0) for every y ∈ P3; however we do not ccomplementary space of P1 as M ≠ P1 + P3.
Example 4.67: Let
Consider
M1 =
1 2
3 4
a a 0 0
a a 0 00 0 0 0
0 0 0 0
-ª º°
« »°« »®« »°« »°¬ ¼¯
ai ∈ Q+ ∪ {0}, 1 ≤
M1 is a semivector subspace of P over S.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 156/344
M2 =
1 2
3 4
0 0 a a0 0 a a
0 0 0 0
0 0 0 0
-ª º°« »°« »®« »°« »°¬ ¼¯
ai ∈ Q+ ∪ {0}, 1 ≤
M2 is a semivector subspace of P over S.Consider
M3 =1 2
3 4
0 0 0 0
0 0 0 0
a a 0 0
a a 0 0
-ª º°« »°
« »®« »°« »°¬ ¼¯
ai ∈ Q+
∪ {0}, 1 ≤
M3 is a semivector subspace of P over S.
Now
0 0 0 0
0 0 0 0
-ª º°« »°
Example 4.68: Let
V =
1 2 3
4 5 6
7 8 9
10 11 12
a a aa a a
a a a
a a a
a a a
-ª º°« »°« »°« »®
« »°« »°« »°¬ ¼¯
ai ∈ Q+ ∪ {0}, 1 ≤ i ≤
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 157/344
13 14 15a a a« »°¬ ¼¯be a semivector space over the semifield S = {0} ∪ Z
So we can define as in case of other spaces comcase of semivector space of polynomials wcoefficients also. We will only illustrate this situatiexamples.
Example 4.69: Let
V = ii
i 0
a x∞
=
-®¯¦ ai = (x1, x2, x3, x4, x5, x6) |
x j ∈ Z+ ∪ {0}; 1 ≤ j ≤ 6}
be a semivector space over the semifield S = Z+ ∪ {0
Consider
M = iia x
∞-®¯¦ ai = (0, 0, 0, x1, x2, x3)
Take
N = ii
i 0
a x∞
=
-®¯¦ ai = (x1, x2, x3, 0,0,0,
with x j ∈ Z+ ∪ {0}; 1 ≤ j ≤ 3} ⊆ V
N is a semivector subspace of V over S. We seei f t M i th th l l t f N
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 158/344
infact M is the orthogonal complement of N That is M⊥ = N and N⊥ = M and M ∩ N = (0).
Example 4.70: Let
V = ii
i 0
a x∞
=
-®¯¦ ai =
1 2 3
4 5 6
7 8 9
10 11 12
13 14 15
d d d
d d dd d d
d d d
d d d
ª º« »« »« »« »« »« »¬ ¼
; d j ∈ Z+ ∪ {
be a semivector space define over the semifield S
Now
W1 =i
ii 0 a x
∞
=
-
®̄¦ xi =
1 2 3
4 5 6
x x x
0 0 0
x x x0 0 0
ª « «
« « « «
1W⊥ = ii
i 0
a x∞
=
-®
¯¦ ai =
1 2 3
4 5 6
0 0 0
y y y
0 0 0
y y y
0 0 0
ª º« »« »« »
« »« »« »¬ ¼
where y j ∈ Z+ ∪ {0}; 1 ≤ j ≤ 6} ⊆ V.
We see W⊥ + W V and W ∩ W⊥ (0)
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 159/344
We see 1W + W1 = V and W1 ∩ 1W = (0).
Suppose
M1 = ii
i 0
a x∞
=
-®¯¦ xi =
1 2 3
0 0 0
0 0 0
0 0 0
x x x
0 0 0
ª º« »« »« »
« »« »« »¬ ¼
with x j ∈ Z+
1 ≤ j ≤ 3} ⊆ V
be another semivector subspace of V; we see M1
orthogonal complement of W1 but however W1 ∩ M1
W1 + M1 ⊆ V. Hence we can have orthogonalsubspaces but they do not serve as the orthogonal cof W1.
Example 4.71: Let
Consider
P1 =i
ii 0 a x
∞
=
-
®̄¦ ai =
1 2
3
d d 0
d 0 00 0 0
ª º« »« »« »¬ ¼
with d j ∈
1 ≤ j ≤ 3} ⊆ V
i t b f V S W th
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 160/344
a semivector subspace of V over S. We see the c
P1 is P2 = ii
i 0
a x∞
=
-®¯¦ ai =
1
2 3
4 5 6
0 0 d
0 d d
d d d
ª º« »« »« »¬ ¼
with d j
1 ≤ j ≤ 6} ⊆ V.
We see P1 + P2 = V and P1 ∩ P2 = {0}. orthogonal complement of P1 and vice versa.
However if
N = ii
i 0
a x∞
=
-®¯¦ ai =
1
2
3
0 0 d
0 0 d
0 0 d
ª º« »« »« »¬ ¼
with d j ∈ Z+ ∪ {0}; 1 ≤ j ≤ 3} ⊆ V
For
T =i
ii 0 a x
∞
=
-
®̄¦ ai = 1
2 3
0 0 0
0 d 0d d 0
ª º« »« »« »¬ ¼
with d j ∈ Z+ ∪ {0}; 1 ≤ j ≤ 3} ⊆ V
is such that T is a semivector subspace of V and
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 161/344
is such that T is a semivector subspace of V andorthogonal with P1 but is not the orthogonal compleas T + P1 ⊂ V. Thus we see there is a differencesemivector subspace orthogonal with a semivector suan orthogonal complement of a semivector subspace.
Now having seen examples of complement subspace and orthogonal complement of a semivectwe now proceed onto give one or two examples of pssum of semivector subspaces.
Example 4.72: Let
V = ii
i 0
a x∞
=
-®¯¦ ai = (x1, x2, …, x8)
where x j ∈ Z+ ∪ {0}; 1 ≤ j ≤ 8}
be a semivector space over the semifield S = Z+ ∪ {0
Consider
W2 = iii 0
a x∞
=-®¯¦ ai = (x1, x2, x3, 0,0,0
with x j ∈ Z+ ∪ {0}; 1 ≤ j ≤ 3} ⊆ V
another semivector subspace of V
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 162/344
another semivector subspace of V.
Take
W3 = ii
i 0
a x∞
=
-®¯¦ ai = (0, 0, d1, 0, 0, d2,
with d j ∈ Z+ ∪ {0}; 1 ≤ j ≤ 2} ⊆ V
another semivector subspace of V over S.
Finally let
W4 = ii
i 0
a x∞
=
-®¯¦ ai = (x1, 0, 0, x2, 0,0, x
with x j ∈ Z+ ∪ {0}; 1 ≤ j ≤ 4} ⊆ V
a semivector subspace of V over S. We see V ⊆
Example 4.73: Let
V = ii
i 0
a x∞
=
-®¯¦ ai =
1
2
3
4
5
6
x
x
xx
x
x
x
ª º« »« »
« »« »« »« »« »« »« »
where x j ∈ Q+ ∪ {0}; 1
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 163/344
7
8
x
x« »« »¬ ¼be a semivector space over the semifield S = Q+ ∪ {0
Take
W1 = ii
i 0
a x∞
=
-®¯¦ ai =
1
2
3
xx
x
0
0
ª º« »« »« »« »« »« »
« »« »¬ ¼
#
where x j ∈ Q+ ∪ {0}; 1 ≤
be a semivector subspace of V over S.
Consider0
0x
ª º
« »« »« »
W3 = ii
i 0
a x∞
=
-®¯¦ ai =
1
2
3
0
0
0
0x
x
x
ª º« »« »« »
« »« »« »« »« »« »« »
where x j ∈ Q+ ∪ {0}
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 164/344
0« »¬ ¼be a semivector subspace of V over S and
W4 = ii
i 0
a x∞
=
-®¯¦ ai =
1
2
3
0
0
00
0
x
x
x
ª º« »« »« »« »« »« »« »« »« »« »« »¬ ¼
where x j ∈ Q+ ∪ {0};
be a semivector subspace of V over S, the semifie
We see Wi ∩ W j = (0); i ≠ j. But V ⊆ W1 +
1 ≤ i, j ≤ 4. Thus V is the pseudo direct sumsubspaces of V over S.
be a semivector space over the semifield S = Z+ ∪ {0
Take
M1 = ii
i 0
a x∞
=
-®¯¦ ai =
1 2 3d d d 0
0 0 0 0
0 0 0 0
0 0 0 0
ª º« »« »« »« »¬ ¼
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 165/344
where d1, d2, d3 ∈ Z+ ∪ {0}}⊆ V,
M2 = ii
i 0
a x∞
=
-®
¯
¦ ai =
1 2
3
0 0 d d
d 0 0 0
0 0 0 0
0 0 0 0
ª º« »« »« »« »¬ ¼
where d1, d2, d3 ∈ Z+ ∪ {0}} ⊆ V,
M3 = ii
i 0
a x∞
=
-®¯¦ ai = 1 2 3
0 0 0 0
d d d 0
0 0 0 0
0 0 0 0
ª º
« »« »« »« »¬ ¼
where d1, d2, d3 ∈ Z+ ∪ {0}}⊆ V,
M5 = ii
i 0
a x∞
=
-®¯¦ ai =
1 2 3
0 0 0
0 0 0
d d d
0 0 0
ª « « «
« ¬
where d1, d2, d3 ∈ Z+ ∪ {0}}⊆ V
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 166/344
M6 = ii
i 0
a x∞
=
-®¯¦ ai =
1 2
0 0 0 0
0 0 0 0
0 d d d
0 0 0 0
ª « « « « ¬
where d1, d2, d3 ∈ Z+ ∪ {0}} ⊆ V
M7 = ii
i 0
a x∞
=
-®
¯¦ ai =
2 3
0 0 0 0
0 0 0 0
0 0 0 dd d 0 0
ª « «
« « ¬
where d1, d2, d3 ∈ Z+ ∪ {0}} ⊆ V
and
-
0 0 0 0
0 0 0 0
ª «
Example 4.75: Let
V = ii
i 0
a x∞
=
-®¯¦ ai =
1 2 3 4 5
7 8 9 10 11
13 14 15 16 17
19 20 21 22 23
d d d d d
d d d d d
d d d d d
d d d d d
ª « « « « ¬
+
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 167/344
where d j ∈ Z ∪ {0}; 1 ≤ j ≤ 24}be a semivector space over the semifield S = Z+ ∪ {0
Consider
P1 =i
ii 0 a x
∞
=
-
®̄¦ ai =
1 2 3 4d d d d 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
ª «
« « « ¬
where d1 , d2, d3, d4 ∈ Z+ ∪ {0}} ⊆ V,
P2 = ii
i 0
a x∞
=
-®¯¦ ai =
1 2
4
0 0 0 d d dd 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
ª « « « « ¬
with d1 , d2, d3, d4 ∈ Z+ ∪ {0}} ⊆ V,
P4 = ii
i 0
a x∞
=
-®¯¦ ai = 1 2
4
0 0 0 0 0
0 0 0 d d
d 0 0 0 0
0 0 0 0 0
ª « « «
« ¬
with d1 , d2, d3, d4 ∈ Z+ ∪ {0}} ⊆ V
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 168/344
P5 = ii
i 0
a x∞
=
-®¯¦ ai =
1 2 3 4
0 0 0 00 0 0 0
d d d d
0 0 0 0
ª « « « « ¬
with d1 , d2, d3, d4 ∈ Z+
∪ {0}} ⊆ V
P6 = ii
i 0
a x∞
=
-®¯¦ ai =
1 2
4
0 0 0 0 0
0 0 0 0 0
0 0 0 d d
d 0 0 0 0
ª « « « « ¬
d1 , d2, d3, d4 ∈ Z+ ∪ {0}} ⊆ V,
P7 = ii
i 0
a x∞
=
-®¯¦ ai =
0 0 0 0
0 0 0 00 0 0 0
ª « « « «
P8 = ii
i 0
a x∞
=
-®¯¦ ai =
1 2 3 4
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 d d d d d
ª « « «
« ¬
with d1 , d2, d3, d4, d5 ∈ Z+ ∪ {0}} ⊆ V
be semivector subspaces of V over S = Z+ ∪ {0}. W
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 169/344
P1 + P2 + P3 + P4 + P5 + P6 + P7 + P8; and Pi ∩ P j ≠ {≤ i, j ≤ 8. Thus V is only a pseudo direct sum ofsubspaces.
&KDSWHU)LYH
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 170/344
1$785$/352'8&721683(50
In this chapter we define the new notion of in supermatrices. Products in supermatrices arfrom usual product on matrices and product on su
Throughout this chapter
SRF = {(a1 a2 a3 | a4 a5 | … | an-1 an) | ai ∈ Q o
collection of 1 × n super row matrices withpartition in it.
1a-ª º°« »
denotes the collection of all m × 1 super column msame type of partition on it.
Sm nF × (m≠n) =
11 12 1n
21 22 2n
m1 m2 mn
a a ... aa a ... a
a a ... a
-ª º°« »
°« »®« »°« »°« »¬ ¼¯
# # # #aij ∈ Q or
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 171/344
1 ≤ i ≤ m; 1 ≤ j ≤ n}
denotes the collection of all m × n super matricestype of partition on it.
Sn nF × =
11 12 1n
21 22 2n
n1 n2 nn
a a ... aa a ... a
a a ... a
-ª º°« »
°« »®« »°« »°« »¬ ¼¯
# # # #aij ∈ Q or Z o
1 ≤ i, j ≤ n}
denotes the collection of n × n super matrices with spartition on it.
We will first illustrate this situation before we p
give any form of algebraic structure on them.
Example 5.2: Let
SRF = {(x1 | x2 x3 | x4 x5 x6 | x7 x8 x9 x10 | x11 x
1 ≤ i ≤ 10}be again a collection of 1 × 10 super row mattype of partition on it.
Example 5.3: Let
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 172/344
P = {(x1 x2 | x3 x4 x5) where xi ∈ Q or R or Z
be again a collection of 1 × 5 super row super mtype.
Now we will see examples of column supsame type.
Example 5.4: Let
SCF =
1
2
3
4
5
6
7
x
x
x
x
x
xx
-ª º
°« »°« »°« »°« »°®« »°« »°« »
°« »°« »¬ ¼°̄
xi ∈ R; 1 ≤ i ≤ 7}
Example 5.5: Let
SCF =
1
2
3
4
5
6
7
x
x
xx
x
x
x
-ª º°« »°« »
°« »°« »°« »°« »®
« »°« »°« »°« »°« »
xi ∈ R; 1 ≤ i ≤ 9}
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 173/344
8
9
x
x
« »°« »°« »°¬ ¼¯
be the 9 × 1 column super matrix of same type.
Example 5.6: Let
SCF =
1
2
3
4
5
6
7
8
a
a
a
aa
a
a
a
-ª º°« »°« »°« »°« »
°« »®« »°« »°« »°« »°« »°« »¬ ¼¯
ai ∈ Q; 1 ≤ i ≤ 8}
be the 8 × 1 column super matrix of same type
Example 5.7: Let
S3 5F × =
1 2 3 4 5
6 7 8 9 10
11 12 13 14 15
a a a a a
a a a a aa a a a a
-ª º°« »°®« »°« »
¬ ¼°̄
ai ∈ Q;
be the 3 × 5 super matrix of same type.
Example 5.8: Let
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 174/344
S7 4F × =
1 2 3 4
5 6 7 8
25 26 27 28
a a a a
a a a a
a a a a
-ª º°« »°« »®« »°« »°« »
¬ ¼¯
# # # #ai ∈ Z; 1
be a 7 × 4 super matrix of same type.
Example 5.9: Let
S6 7F × =
1 2 3 4 5 6
7 8 9 10 11 12
31 32 33 34 35 36
a a a a a a
a a a a a a
a a a a a aa a a a a a
-ª º°« »°« »°« »®
« »°« »°« »°¬ ¼¯
# # # # # # ai ∈
Example 5.10: Let
S4 4F × = M =
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
a a a a
a a a aa a a a
a a a a
-ª º°« »°« »®« »°« »°¬ ¼¯
ai ∈ Q; 1 ≤
be a square super matrix of same type.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 175/344
Example 5.11: Let
S4 4F × =
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
a a a a
a a a a
a a a aa a a a
-ª º°« »°« »®
« »°« »°¬ ¼¯
ai ∈ Q; 1 ≤ i ≤
be a square super matrix of same type.
Example 5.12:Let
S4 4F × =
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
a a a a
a a a a
a a a a
a a a a
-ª º°« »°« »®« »°« »
°¬ ¼¯
ai ∈ Q; 1 ≤ i ≤
Now we can define S S SC R m nF ,F ,F × (m ≠ n) and
matrix addition and natural product ×n.
Under usual matrix addition S S SC R m nF ,F ,F × (m
are abelian (commutative) groups.
How under natural product S S SC R m nF ,F ,F × (m ≠
semigroups with unit
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 176/344
semigroups with unit.
Suppose
x = (x1 x2 x3 | x4 x5 | x6 x7) and y = (y1 y2 y3 |be two super row matrices of same type
x + y = (x1 + y1, x2 + y2, x3 + y3 | x4 + y4, x5 ++ y6 x7 + y7). Thus S
RF is closed und
Likewise if x =
1
2
3
4
5
6
7
x
xx
x
x
x
x
ª º
« »« »« »« »« »« »« »« »« »« »« »
and y =
1
2
3
4
5
6
7
y
yy
y
y
y
y
ª º
« »« »« »« »« »« »« »« »« »« »« »
are two
matrices of same type then x + y =
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
x y
x y
x y
x yx y
x y
x y
x y
+ª º« »+« »« »+
« »+« »« »+« »
+« »« »+« »
+« »¬ ¼
.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 177/344
¬ ¼
We see SCF is closed under ‘+’ and infact a group
Consider x =
1 2 3 4 5 6
8 9 10 11 12 13 1
15 16 17 18 19 20 2
22 23 24 25 26 27 2
29 30 31 32 33 34 3
a a a a a a aa a a a a a a
a a a a a a a
a a a a a a a
a a a a a a a
ª « « « « « «
¬
y =
1 2 3 4 5 6 7
8 9 10 11 12 13 14
15 16 17 18 19 20 21
22 23 24 25 26 27 28
b b b b b b b
b b b b b b b
b b b b b b b
b b b b b b b
ª º« « «
« « «
Now x + y =
1 1 2 9 3 3 4 4 5 5
8 8 9 9 10 10 11 11 12 12
15 15 16 16 17 17 18 18 19 19
22 22 23 23 24 24 25 25 26 26
29 29 30 30 31 31
a b a b a b a b a b
a b a b a b a b a b
a b a b a b a b a b
a b a b a b a b a b
a b a b a b a
+ + + + +
+ + + + +
+ + + + +
+ + + + +
+ + + 32 32 33 33b a b
ª
« « « « « « + + ¬
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 178/344
is in S
5 7F × .
Thus addition can be performed on Sm nF × (m
Sm nF × is a group under addition.
Now we give examples of addition of squareSn nF × .
Let x =
1 2 3 4 5
6 7 8 9 10
11 12 13 14 15
16 17 18 19 20
21 22 23 24 25
a a a a a
a a a a aa a a a a
a a a a a
a a a a a
ª º« « « « « « ¬ ¼
1 2 3 4 5b b b b b
b b b b bª «
x+y =
1 1 2 2 3 3 4 4
6 6 7 7 8 8 9 9
11 11 12 12 13 13 14 14
16 16 17 17 18 18 19 19
21 21 22 22 23 23 24 24
a b a b a b a b
a b a b a b a b
a b a b a b a b
a b a b a b a ba b a b a b a b
+ + + + ª « + + + + « « + + + +
« + + + + « « + + + + ¬
∈ S5 5F × .
Infact S
5 5F
×is a group under addition.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 179/344
Now we proceed onto define natural prodS S SC R n mF ,F ,F × (n ≠ m) and S
n nF × .
Consider x = (a1 a2 a3 | a4 a5 | a6 a7 a8 a9) and y =
b5 | b6 b7 b8 b9) ∈ SRF .
x ×n y = (a1b1 a2b2 a3b3 | a4b4 a5b5 | a6b6 a7b7 a8b8
SRF under product is a semigroup infact S
RF has z
under natural product ×n.
Suppose x = (0 0 0 | 2 1 0 | 92 | 3) and y = (3 9 0 | 0) be in S
RF . x ×n y = (0 0 0| 0 0 0 | 0 0 | 0). Thushas zero divisors.
2
03
ª º« »« »« »
0
10
ª º« »« »« »
we see under the natural product x ×n y
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 180/344
Take S3 5F × , S
3 5F × under natural product, t
divisors. Infact S3 5F × under natural product ×n is a
Consider x =
9 0 2 0 1
0 1 0 5 01 0 0 2 0
ª º« »
« »« »¬ ¼an
y =
0 7 0 8 0
9 0 2 0 7
0 7 9 0 2
ª º« »« »
« »¬ ¼
in S3 5F × ;
Consider x =
7 8 0 9 4 2
0 1 2 5 7 8
1 2 3 0 1 0
5 7 0 9 2 01 2 3 0 2 3
0 8 7 0 5 4
ª º« »« »« »
« »« »« »« »« »¬ ¼
and
0 0 9 0 0 0
7 0 0 0 0 0
ª º
« »« »
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 181/344
y =
7 0 0 0 0 0
0 0 0 6 0 8
0 0 6 0 0 2
0 0 0 6 0 0
5 0 0 7 0 0
« »« »« »« »« »« »« »
« »¬ ¼
∈ S6 6F × .
x ×n y =
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 00 0 0 0 0 0
0 0 0 0 0 0
ª º« »« »« »« »« »« »« »« »¬ ¼
∈ S6 6F × .
Thus Sn nF × is a semigroup under ×n and has zero
ideals. We will now give the following theorems thwhich are simple.
THEOREM 5.2:
S
C F =
−
-ª º°« »
°« »°« »°« »°®« »°« »°« »°« »
°« »°¬ ¼¯
#
#
1
2
3
m 1
m
x
x x
x
x
| xi ∈ Q or R or C or Z; 1
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 182/344
°« »°¬ ¼¯is the group under ‘+’.
THEOREM 5.3 :
×
S
3 3F (m ≠ n) =
-ª º°« »°« »®« »°« »°« »¬ ¼¯
# # #
11 12 1n
21 22 2n
m1 m2 mn
a a ... aa a ... a
a a ... a
| aij ∈ Q o
1 ≤ i ≤ m; 1 ≤ j ≤ n}
is a group under ‘+’.
THEOREM 5.4: ( ×
S
n nF , +) is a group.
THEOREM 5.5: ( S
RF , × n) is a semigroup and ha
THEOREM 5.8: ( ×
S
n nF , × n) is a commutative semigro
zero divisors units and ideals.
We will now give examples of zero divisors unitof S
m nF × (m ≠ n), S S SC R n mF ,F ,F × (n ≠ m) and S
n nF × .
Example 5.14: Let
SRF = {(x1 | x2 x3 | x4) where xi ∈ Z; 1 ≤ i ≤ 4
b t ti i d t l d t
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 183/344
be a commutative semigroup under natural product. is the unit of S
RF under ×n.
P = {(x1 | x2 x3 | x4) | xi ∈ 3Z; 1 ≤ i ≤ 4} ⊆ SRF is
SRF .
Infact SRF has infinite number of ideals under
product ×n. Further SRF has zero divisors.
Consider x = (x1 | 0 0 | x2) ∈ SRF , y = (0 | y1 y2 |
such that x ×n y = (0 | 0 0 | 0). Also P = (x1 | 0 0 | x2
≤ i ≤ 2} ⊆ SRF is also an ideal.
Example 5.15: LetS
RF = {(x1 | x2 x3 | x4 x5 x6) where xi ∈ Q; 1 ≤ ibe a semigroup under ×n.
T = {(a1 | a2 a3 | 0 0 0) | ai ∈ Z, 1 ≤ i ≤ 3}
only a subsemigroup of SRF and is not an ideal o
has subsemigroups which are not ideals.
Take M = {(a | b c | 0 0 0) | a, b, c ∈ Q} ⊆
N = {(0 | 0 0 | a b c) | a, b, c ∈ Q} ⊆
M ×n N = (0 | 0 0 | 0 0 0) or M ∩ N = (0 | 0 0
SRF = M + N.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 184/344
R
Suppose M1 = {(a | b 0 | 0 0 0) | a, b ∈ Q} ⊆
N1 = {(0 | 0 0 | a b 0) | a, b ∈ Q} ⊆ SRF
M1 ×n N1 = {(0 | 0 0 | 0 0 0)|. Also M1 ∩ N1
0)} but N1 + M1 ≠⊂ S
RF ; and N1 + M1 ≠ SRF . W
special properties are enjoyed by M and N thatcase M1 and N1.
Now we give an example in case of ( SCF , ×n).
Example 5.16: Let
1
2
aa
-ª º°« »°« »
Consider
P =
1
2
3
4
a
aa
a
0
0
0
-ª º°« »
°« »°« »°« »°®« »°« »°« »°« »
°« »¬ ¼°̄
ai ∈ Q, 1 ≤ i ≤ 4} ⊆ SCF
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 185/344
is a subsemigroup of SCF and is not an ideal of S
CF .
Take
M =
1
2
3
00
0
0
aa
a
-ª º°« »°« »°« »°« »°®« »°« »°« »°« »°« »
¬ ¼°̄
ai ∈ Z, 1 ≤ i ≤ 3} ⊆ SCF ,
M is a subsemigroup of SCF .
Clearly if x ∈ P and y ∈ M then
Now take
S =
1
2
3
4
5
6
7
a
a
aa
a
a
a
-ª º°« »°« »
°« »°« »°®« »°« »°« »°« »°« »
¬ ¼°¯
ai ∈ Z, 1 ≤ i ≤ 7} ⊆ CF
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 186/344
S is a subsemigroup of SCF but is not an ideal of F
Suppose
J =
1
2
3
4
a0
0
0
aa
a
-ª º°« »°« »°« »°« »°®« »°« »°« »°« »°« »
¬ ¼°̄
ai ∈ Q, 1 ≤ i ≤ 4} ⊆ F
J is a subsemigroup as well as an ideal of SCF .
Now we give yet another example.
Example 5.17: Let
SCF =
1
2
3
4
5
6
7
a
aa
a
a
a
a
-ª º°« »
°« »°« »°« »®« »°« »°« »°« »
°« »¬ ¼¯
ai ∈ Q, 1 ≤ i ≤ 7}
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 187/344
be a semigroup under natural product ×n.
Take
P =
1
2
3
0
a0
a
0
a
0
-ª º
°« »°« »°« »°« »®« »°« »°« »°« »°« »¬ ¼¯
ai ∈ Q, 1 ≤ i ≤ 3} ⊆ SCF
is a subsemigroup as well as an ideal of SCF .
Take
10a-ª º°« »
°« »
Now take
T =
1
2
3
4
a
0a
0
a
0
a
-ª º°« »
°« »°« »°« »®« »°« »°« »°« »
°« »¬ ¼¯
ai ∈ Q, 1 ≤ i ≤ 4} ⊆
S
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 188/344
T is a subsemigroup of SCF also an ideal of
every x ∈ P and for every y ∈ T, x ×n y = (0).
Now we give examples of zero divisors and
(n ≠ m).
Example 5.18: Let
S5 3F × =
1 2 3 4 5
6 7 8 9 10
11 12 13 14 15
a a a a a
a a a a aa a a a a
-ª º°« »®« »°« »¬ ¼¯
ai ∈ Q,
be a semigroup of 5 × 3 super matrices multiplication.
1 2 3 4 5ª º« »
0 1ª «
then x ×n y =
0 2 6 12 25
81 0 7 18 20
0 2 6 7 2
ª º« »« »« »¬ ¼
.
Now consider P =
a b c d e
0 0 0 0 0
0 0 0 0 0
-ª º°« »®« »°« »¬ ¼¯
a, b, c, d
S
5 3
F×
; P is an ideal of S
5 3
F×
.
ª º
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 189/344
We see for a =
0 0 0 0 0
x y a b z
0 0 c d m
ª º« »« »« »¬ ¼
∈ S5 3F × is suc
a ×n x =
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
ª º« »« »« »¬ ¼
for every x ∈ P
Take
M =
0 0 e f 0
a b 0 0 g
c d 0 0 h
-ª º°« »®« »°« »¬ ¼¯
a, b, c, d, e, f, g, h ∈ Z
clearly M is only a subring of S5 3F ; and is not an idea
Example 5.19: Let
S7 3F × =
1 2 3
4 5 6
7 8 9
10 11 12
13 14 15
16 17 18
a a aa a a
a a a
a a a
a a a
a a a
-ª º°« »°« »°« »°« »°®« »°« »
°« »°« »°« »
ai ∈ Q, 1 ≤
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 190/344
19 20 21a a a°« »¬ ¼°̄
be a 7 × 3 super matrix semigroup under natural
Consider
P =1 2 3
4 5 6
0 0 0
0 0 0a a a
0 0 0
0 0 0
0 0 0
a a a
-ª º
°« »°« »°« »°« »°®« »°« »°« »
°« »°« »¬ ¼°̄
ai ∈ Z, 1 ≤ i ≤ 6} ⊆
Now take
x =
1 2 3
4 5 6
7 8 9
10 11 12
13 14 15
a a a
a a a
0 0 0
a a a
a a a
a a a
0 0 0
ª º« »
« »« »« »« »« »« »« »« »¬ ¼
∈ S7 3F × .
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 191/344
Clearly x ×n p = (0) for every p ∈ P.
Thus we have a collection of zero divisors in theunder natural product.
Now consider the set
T =
1 2 3
4 5 6
7 8 9
10 11 12
a a a
0 0 0a a a
a a a
0 0 0
0 0 0
a a a
-ª º°« »°« »°« »°« »°®« »°« »°« »°« »°« »¬ ¼°̄
ai ∈ Q, 1 ≤ i ≤ 12} ⊆
Further
m =
1 2 3
4 5 6
7 8 9
0 0 0
a a a
0 0 00 0 0
a a a
a a a
0 0 0
ª º« »« »
« »« »« »« »« »« »« »¬ ¼
∈ S7 3F ×
is such that m ×n t = (0) for every t ∈ T. Thuszero divisors and has ideals
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 192/344
zero divisors and has ideals.
Example 5.20: Let
S3 7F × =
1 2 3 4 5 6
8 9 10 11 12 13
15 16 17 18 19 20
a a a a a aa a a a a a
a a a a a a
-ª °« ®« °« ¬ ¯
ai ∈ Q, 1 ≤ i ≤ 21}
be 3 × 7 matrix semigroup under natural product.
Take
P =
1 4 7
2 5 8
3 6 9
0 a 0 a 0 a 0
0 a 0 a 0 a 00 a 0 a 0 a 0
-ª º°« »
®« »°« »¬ ¼¯ai ∈ Q, 1 ≤
Thus S
3 7F × has several zero divisors.
Take
Y =1 2 3 4 5 6 7
8 9 10 11 12 13 14
15 16 17 18 19 20 21
a a a a a a a
a a a a a a a
a a a a a a a
-ª º°« »®« »°« »¬ ¼¯
ai ∈ Q,
⊆
S
3 7F × , Y is only a subsemigroup and not an ideal of
E l 5 21 L t
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 193/344
Example 5.21: Let
M =
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
a a a a
a a a a
a a a a
a a a a
-ª º°« »
°« »®« »°« »°« »¬ ¼¯
ai ∈ Q, 1 ≤ i ≤
be a semigroup under natural product ×n.
Consider
P =
1 2
3 5
4 6
7 8
0 a a 0
a 0 0 a
a 0 0 a0 a a 0
-ª º°« »°« »®« »°« »°« »¬ ¼¯
ai ∈ Z, 1 ≤ i ≤ 8} ⊆ S4F
Let
X =
1 2
5 6
7 8
3 4
a 0 0 a
0 a a 0
0 a a 0
a 0 0 a
-ª º°« »
°« »®« »°« »°« »¬ ¼¯
ai ∈ Q, 1 ≤ i ≤ 8
X is an ideal of S4 4F × . Further every x ∈ P and m
(0). Thus S4 4F × has zero divisors and subsemigrnot ideals.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 194/344
Now consider another example.
Example 5.22: Let
S3 3F × =
1 2 3
4 5 6
7 8 9
a a a
a a a
a a a
-ª º°« »°®« »°« »
¬ ¼°̄
ai ∈ Z, 1 ≤ i
be a 3 × 3 super matrix semigroup under naturaimportant to observe S
3 3F × is not compatible wiproduct. Also no type of product on square supbe defined on elements in S
3 3F × .
Take
Take
M =
1 2 3
4 5 6
a a a
0 0 0a a a
-ª º°« »°
®« »°« »¬ ¼°̄
ai ∈ Z, 1 ≤ i ≤ 6} ⊆ F
M is a subsemigroup as well as an ideal of S3 3F × .
every x ∈ X and m ∈ M, x ×n m = (0).
Now we describe the unit element of S S SC R mF ,F ,F
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 195/344
and Sn nF × .
In SCF ,
1
1
1
1
1
1
ª º« »« »« »« »« »
« »« »« »« »¬ ¼
#
acts as the supercolumn unit under
product ×n.
For SRF ; (1 1 | 1 1 1 | 1 … | 1 1) acts as the sup
l d h l d
ForS7 3F × ;
1 1 1
1 1 1
1 1 1
1 1 11 1 1
1 1 1
1 1 1
ª º« »« »« »« »« »« »« »« »« »¬ ¼
acts as the super 7 × 3
natural product ×n.
1 1 1 1ª º
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 196/344
For S4 4F × ,
1 1 1 1
1 1 1 1
1 1 1 1
ª º« »« »« »« »
« »¬ ¼
acts as the 4 × 3 s
product.
Take x = (1 | 1 1 | 1 1 1 | 1 1) (7 | 3 2 | 5 7 -1= (7 | 3 2 | 5 7 -1 | 2 0).
Likewise for x =
3
2
1
0
31
ª º« »« »« »−« »« »
« »« »« »
,
1
1
1
1
11
ª º« »« »« »« »« »
« »« »« »
act as the multip
8 × 1 identity for,
3
2
1
03
1
7
0
2
ª º« »« »« »−« »
« »« »« »« »« »« »« »
« »¬ ¼
×n
1
1
1
11
1
1
1
1
ª º« »« »« »« »
« »« »« »« »« »« »« »
« »¬ ¼
=
3
2
1
03
1
7
0
2
ª º« »« »« »−« »
« »« »« »« »« »« »« »
« »¬ ¼
.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 197/344
For x =
3 7 2 5 1 0 1 7 0 8
0 1 2 3 4 5 6 7 8 9
0 3 4 0 1 0 7 0 1
4 0 2 1 0 2 0 4 0 0
−ª « « « « « ¬
I =
1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1
ª º« »
« »« »« »« »¬ ¼
acts as the super identity under ×n. For x ×n I = I ×n x
Consider
Now having seen how the units look like w
onto see how inverse of an element look under ×n.
Let
x =
7 3 1
1 2 9
8 5 1
4 7 2
−ª º« »« »« »« »¬ ¼
,
if t k it t i ith f Q f R d
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 198/344
if x takes its entries either from Q or from R andis zero then alone inverse exists otherwise inversexist.
Take y =
1/ 7 1/ 3 11 1/ 2 1/ 9
1/8 1/5 1
1/ 4 1/ 7 1/ 2
−ª º« »« »« »« »¬ ¼
then we
x ×n y =
1 1 1
1 1 1
1 1 1
1 1 1
ª º« »« »« »« »¬ ¼
.
Let
x ×n y =
1 1 1
1 1 1
1 1 1
1 1 1
ª º« »« »« »« »
¬ ¼
.
Consider x = (1/8 | 7 5 | 3 2 4 –1) then the inver= (8 | 1/7 1/5 | 1/3 1/2 1/4 –1) we x ×n y = (1 | 1 1 | 1
Consider
8 1ª º« »
1/8 1ª º« »
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 199/344
x =
3 5
1 1/ 7
8 4
1 1
3 2
« »« »« »−« »−« »
« »−« »
−« »¬ ¼
then y =
1/ 3 1/ 5
1 7
1/8 1/ 4
1 1
1/ 3 1/ 2
« »« »« »−« »−« »
« »−« »
−« »¬ ¼
is such that
x ×n y =
1 1
1 1
1 1
1 1
1 1
1 1
ª º
« »« »« »« »« »« »« »
« »¬ ¼
.
THEOREM 5.10: Let S
C F (or S
RF or ×
S
m nF (m ≠the super matrix semigroup under natural produ
from Q or R. Every super matrix M in which n
takes 0 has inverse.
The proof of this theorem is also left as an
reader.
C id (1 1 | 1 1 1 | 1 1) SF {
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 200/344
Consider x = (1 –1 | 1 1 –1 | –1 –1) ∈ SRF = {
a6 a7) | ai ∈ Z, 1 ≤ i ≤ 7}; clearly x = (1 –1 | 1 1 as its inverse that is x ×n x = (1 1 | 1 1 1 | 1 1).
Consider
y =
1
1
1
1
1
1
1
11
−ª º« »« »« »−
« »« »« »−« »« »« »−« »
« »« »¬ ¼
∈ SCF =
1
2
3
4
5
6
7
8
a
a
a
a
a
a
a
aa
-ª º°« »°« »°« »°« »°« »°« »®
« »°« »°« »°« »°« »°« »°¬ ¼¯
where ai ∈ Z; 1
Now
y2 =
1
1
11
1
1
1
11
ª º« »« »« »« »« »« »« »« »« »« »
« »« »¬ ¼
;
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 201/344
all y whose entries are from Z \ {1, -1} does not hunder natural product.
Take
y =
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
− −ª º« »− −« »« »− − −« »
−« »
« »− − −¬ ¼
we see y2 =
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
ª º« »« »« »
« »« »« »
.
Consider y =
1
0
2
35
7
0
2
14
−ª º« »« »« »« »
« »« »−« »« »« »« »« »
« »« »« »¬ ¼
;
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 202/344
¬ ¼
clearly y-1 does not exist.
Consider
x =7 1 0 2 3 4 0 3
0 2 1 0 7 0 1 0
− ª « ¬
Clearly x-1 does not exist.
Now we have seen inverse of a super matriproduct and the condition under which the inverse
Now we proceed onto discuss the operation ‘
or Sm nF × (m ≠ n) or S
n nF × , which is stated atheorems.
THEOREM 5.14: ( ×
S
n nF , +) is an additive abelian gro
square matrices.
We can define subgroups. All subgroups arethese groups are abelian. We will just give some exam
Example 5.23: LetSRF = {(a1 a2 a3 a4 | a5 a6 | a7 a8 | a9) | ai ∈ Q; 1
be an abelian group of super row matrices under addit
Example 5.24: Let
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 203/344
S2 3F × =
1 2 3
4 5 6
a a a
a a a
-ª º°« »®« »°¬ ¼¯
where ai ∈ Q; 1 ≤ i ≤
be an additive abelian group of 2 × 3 super matrices.
Example 5.25: Let
S
CF =
1
2
3
4
5
6
7
a
aa
a
a
a
a
-ª º
°« »°« »°« »°« »°« »°« »°« »°
« »®« »°
« »°
ai ∈ Q, 1 ≤ i ≤ 11}
Example 5.26: Let
S4 4F × =
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
a a a a
a a a a
a a a a
a a a a
-ª º°« »°
« »®« »°« »°¬ ¼¯
ai ∈ R, 1 ≤
be an additive abelian group of 4 × 4 super matric
Now we can define {
S
RF , +, ×n} as the rinmatrices, ( S
CF , +, ×n) as the ring of super co
{ SF (m ≠ n) × +} is the ring of super m ×
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 204/344
{ m nF × (m ≠ n), ×n, +} is the ring of super m ×
{ Sn nF × , ×n, +} be the ring of super n × n matrices.
We describe properties associated with them.
Example 5.27: Let
SCF =
1
2
3
4
5
6
7
8
a
a
aa
a
a
a
a
-ª º°« »°« »°
« »°« »°« »®« »°« »°« »°« »°« »°« »¬ ¼¯
ai ∈ Q, 1 ≤ i ≤ 8}
Example 5.29: Let
S3 4F × =
1 4 7 10
2 5 8 11
3 6 9 12
a a a a
a a a a
a a a a
-ª º°« »®
« »°« »¬ ¼¯
ai ∈ Q, 1 ≤ i ≤ 12,
be the ring of 3 × 4 supermatrices.
Example 5.30: Let
1 2 3 4a a a a-ª º°« »°
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 205/344
S3 4F × = 5 6 7 8
9 10 11 12
13 14 15 16
a a a a
a a a a
a a a a
°« »°« »®« »°« »°
¬ ¼¯
ai ∈ Z, 1 ≤ i ≤
be the ring of square supermatrices.
Example 5.31: Let
S
9 3
F×
=
1 2 3
4 5 6
7 8 9
10 11 12
13 14 15
16 17 18
a a aa a a
a a a
a a a
a a a
a a a
-ª º°« »°« »°« »°« »°« »°« »
®« »°« »°
ai ∈ Q, 1 ≤ i ≤ 27, +
Example 5.32: Let
S3 10F × =
1 2 3 4 5 6 7
11 12 13 14 15 16 17
21 22 23 24 25 26 27
a a a a a a a a
a a a a a a a a
a a a a a a a a
-ª °« ®
« °« ¬ ¯
ai ∈ Q, 1 ≤ i ≤ 30, +, ×n}
be a ring of super row vectors.
All these rings are commutative have zero dunit. However we will give examples of ring of
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 206/344
g p gwhich have no unit.
Example 5.33: Let
SRF = {(x1 | x2 x3 x4 | x5 x6 | x7 x8 x9 x10) | x
1 ≤ i ≤ 10, +, ×n}
be the ring of super row matrices. Clearly SRF d
the unit (1 | 1 1 1 | 1 1 | 1 1 1 1).
Example 5.34: Let
1
2
3
a
a
a
-ª º°« »
°« »°« »°« »
be the ring of super column matrices. Clearly this super identity.
Example 5.35: Let
S3 4F × =
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
a a a a
a a a a
a a a a
a a a a
-ª º°« »°« »®« »°« »°¬ ¼¯
ai ∈ 10Z; 1 ≤ i ≤ 1
be a ring of 4 × 4 super matrices
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 207/344
be a ring of 4 × 4 super matrices.
Clearly the super unit
1 1 1 1
1 1 1 11 1 1 1
1 1 1 1
ª º« »« »« »« »¬ ¼
∉ M.
Example 5.36: Let S3 4F × =
1 2 3 4 5 6 7 8 9
13 14 15 16 17 18 19 20 21
25 26 27 28 29 30 31 32 33
x x x x x x x x x
x x x x x x x x x
x x x x x x x x x
-ª °« ®« °« ¬ ¯
xi ∈ 5Z; 1 ≤ i ≤ 36, +, ×}
Example 5.37: Let
V =
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
17 18 19 20
21 22 23 24
25 26 27 28
29 30 31 32
a a a a
a a a a
a a a a
a a a a
a a a a
a a a a
a a a aa a a a
-ª º°« »°
« »°« »°« »°« »®« »°« »°« »°« »°« »°« »¬ ¼¯
a j ∈ 15Z; 1 ≤
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 208/344
be a ring of super column vectors which has no un
Now we proceed onto study super matrix str∪ {0} or Q+ ∪ {0} or Z+ ∪ {0}.
Let RS+ = {(x1 x2 x3 | x4 … | xn-1 xn) | xi ∈ R+
{0} or Z+ ∪ {0}} denotes the collection ofmatrices of same type from R+ ∪ {0} or Q+ ∪ {
This notation will be used throughout this book.
CS+ =
1
2
3
a
a
a
-ª º°« »°« »°« »®
« »°« »°#
a j ∈ Z+ ∪ {0} or R+ ∪
m nS+ × (m ≠ n) =
11 12 1n
21 22 2n
m1 m2 mn
a a ... a
a a ... a
a a ... a
-ª º°« »°
« »®« »°« »°« »¬ ¼¯
# # # aij ∈ Q
or Z+ ∪ {0} or R+ ∪ {0}; 1 ≤ i ≤ m, 1 ≤ j
denotes the collection of all m × n super matrices owith entries from Q+ ∪ {0} or Z+ ∪ {0} or R+ ∪ {0}
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 209/344
n nS+× =
11 12 1n
21 22 2n
n1 n2 nn
a a ... a
a a ... a
a a ... a
-ª º°« »°
« »®« »°« »°« »¬ ¼¯
# # # aij ∈ Z+ ∪ {0
or Q+ ∪ {0} or R+ ∪ {0}; 1 ≤ i, j ≤ n}
denotes the collection of all n × n super matrices owith entries from R+ ∪ {0} or Q+ ∪ {0} or Z+ ∪ {0}
We will first illustrate these situations by some ex
Example 5.38: Let
Example 5.39: Let RS+ =
1 2 3 4 5 6 7 8
10 11 12 13 14 15 16 1
19 20 21 22 23 25 25 2
a a a a a a a a
a a a a a a a aa a a a a a a a
-ª °«
®« °« ¬ ¯
ai ∈ Z+ ∪ {0}; 1 ≤ i ≤ 27}
be the set of all super row vectors of same tyfrom Z+ ∪ {0}.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 210/344
Example 5.40: Let
CS+ =
1
2
3
4
5
6
7
8
9
10
11
aa
a
a
a
aa
a
a
a
a
-ª º°« »°« »°« »°« »°« »°« »°
« »°« »®« »°« »°« »°« »°« »°« »°« »°« »°¬ ¼¯
ai ∈ Q+ ∪ {0}; 1 ≤ i ≤
Example 5.41: Let
CS+ =
1 2 3
4 5 6
7 8 9
10 11 12
13 14 15
16 17 18
19 20 21
22 23 24
25 26 27
a a a
a a a
a a a
a a a
a a a
a a a
a a aa a a
a a a
-ª º°« »°
« »°« »°« »°« »°« »°« »°« »°®« »°« »°« »°« »°« »
ai ∈ R+ ∪ {0}; 1 ≤ i ≤
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 211/344
28 29 30
31 32 33
34 35 36
a a a
a a a
a a a
°« »°« »°« »°« »
°« »¬ ¼°̄
denote the collection of all super column vectors owith entries from R+ ∪ {0}.
Example 5.42: Let
3 4S+
× =1 2 3 4
5 6 7 8
9 10 11 12
a a a a
a a a a
a a a a
-ª º°« »°®« »
°« »¬ ¼°̄
ai ∈ R+ ∪ {0}; 1 ≤
Example 5.43: Let
5 5S+
× =
1 2 3 4 5
6 7 8 9 10
11 12 13 14 15
16 17 18 19 20
21 22 23 24 25
a a a a aa a a a a
a a a a a
a a a a a
a a a a a
-ª º°« »°« »°« »®
« »°« »°« »°¬ ¼
¯
ai ∈ R+ ∪ {
be the collection of 5 × 5 super matrices of entries from R+ ∪ {0}
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 212/344
entries from R ∪ {0}.
Now we proceed onto give all possible alge
on CF+ , RF+ or m nF+× (m ≠ n) and n nF+× .
Consider CF+ the collection of all super colu
same type with entries from Q+ ∪ {0} or R+ ∪ {
CS+ is a semigroup under ‘+’ usual addition. I
additive identity and ( CS+ , +) is a commutatLikewise RF+ or m nF+
× (m ≠ n) and n nF+
×
semigroups with respect to addition. Infact monoids.
We will illustrate this by some examples.
Example 5.44: Let
CS+ =
1
2
3
4
5
6
7
8
a
a
a
a
a
a
aa
-ª º°« »°
« »°« »°« »°« »®« »°« »°« »°« »°« »°« »¬ ¼¯
ai ∈ Z+ ∪ {0}; 1 ≤ i ≤ 8}
be a commutative semigroup of super column m
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 213/344
g p pentries from Z+ ∪ {0}.
Example 5.45: Let
m nS+
× =
1 8 15 22
2 9 16 23
3 10 17 24
4 11 18 25
5 12 19 26
6 13 20 27
7 14 21 28
a a a a
a a a a
a a a aa a a a
a a a a
a a a a
a a a a
-ª º°« »°« »°
« »°« »°®« »°« »°« »°« »°« »
¬ ¼°̄
ai ∈ Q+ ∪ {0}; 1 ≤
Example 5.46: Let
4 4S+
× =
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
a a a a
a a a a
a a a a
a a a a
-ª º°« »°
« »®« »°« »°« »¬ ¼¯
ai ∈ R+
∪ {0}
be the semigroup of super 4 × 4 super matric
from R
+
∪ {0}. Example 5.47: Let
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 214/344
3 9S+
× =1 4 7 10 13 16 19
2 5 8 11 14 17 20
3 6 9 12 15 18 21
a a a a a a a
a a a a a a a
a a a a a a a
-ª °« ®« °« ¬ ¯
ai ∈ Q+ ∪ {0}; 1 ≤ i ≤ 27}
be the semigroup of super row vector under
elements from Q+ ∪ {0}.
Example 5.48: Let
1 2 3 4
5 6 7 8
9 10 11 12
a a a a
a a a a
a a a a
-ª º°« »°« »
°« »°« »°
be the semigroup of super column vector under additio
Now we proceed onto define natural produ
RS+ , n nS+
× and m nS+
× (m ≠ n). Clearly CS+ , RS+ , n mS+
×
m mS+× are semigroups under the natural product ×n.
Depending on the set from which they take their will be semigroups with multiplicative identity or othe
We will illustrate this situation by some examples
Example 5.49: Let RS+ = {(a1 | a2 a3 | a4 a5 a6 | a7 a8
a12) | ai ∈ Z+ ∪ {0}, 1 ≤ i ≤ 12} be a semigroup o
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 215/344
a12) | ai ∈ Z ∪ {0}, 1 ≤ i ≤ 12} be a semigroup omatrices under the natural product ×n.
Example 5.50: Let
CS+ =
1
2
3
4
5
6
7
8
9
x
x
x
xx
x
x
x
x
-ª º°« »°« »°« »°
« »°« »°« »°« »°
« »®« »°« »°« »
°« »°« »°
xi ∈ Z+ ∪ {0}, 1 ≤ i ≤ 11}
Example 5.51: Let
CS+ = 1 2 3 4 5 6 7 8
11 12 13 14 15 16 17 1
a a a a a a a a
a a a a a a a a
-ª °®«
¬ °̄
ai ∈ R+ ∪ {0}, 1 ≤ i ≤ 20}
be the semigroup of super row vector under the ×n. 2 10S+
× has identity elements, units and zero div
Example 5.52: Let
1 2 3a a a-ª º°« »
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 216/344
8 3S+
× =
4 5 6
7 8 9
10 11 12
13 14 15
16 17 18
19 20 21
22 23 24
a a a
a a a
a a aa a a
a a a
a a a
a a a
°« »°« »°« »
°« »°« »®« »°« »°« »°« »°« »°« »
¬ ¼¯
ai ∈ Z+ ∪ {0}, 1
be a semigroup of super column vectors under ×n.
1 1 1
1 1 1
ª º« »« »
Example 5.53: Let
2 2S
+
× =
1 2
3 4
a a
a a
-ª º°
« »®« »°¬ ¼¯ ai ∈ Z
+
∪ {0}, 1 ≤ i ≤ 2
be the semigroup of super square matrices under natu
×n. Clearly1 1
1 1
ª º« »« »¬ ¼
is the identity element of 2 2S+
× ,
units, that is no element in 2 2S+
× has inverse. Furth
zero divisors. For take x =1 2
0 0
a a
ª º« »« »¬ ¼
in 2 2S+
×
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 217/344
1 2a a
0 0
ª º« »
« »¬ ¼
and y3 =10 a
0 0
ª º« »
« »¬ ¼
are all zero divisors in S
Example 5.54: Let
8 4S+
× =
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
17 18 19 20
21 22 23 24
25 26 27 28
a a a a
a a a a
a a a a
a a a a
a a a a
a a a a
a a a a
-ª º°« »
°« »°« »°« »°« »®« »°« »°« »°
« »°« »°« »
ai ∈ Q+ ∪ {0}, 1
8 4S+× is a semigroup with unit I =
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 11 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
ª º« »« »« »« »
« »« »« »« »« »« »« »¬ ¼
an
divisors and inverses. Now we can find ideals, zero divisors and units in semigroups under the × These will be only illustrated by some examp
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 218/344
×n. These will be only illustrated by some examp
Example 5.55: Let
CS+ =
1 2 3
4 5 6
7 8 9
10 11 12
13 14 15
16 17 18
19 20 21
22 23 24
25 26 27
a a a
a a a
a a a
a a aa a a
a a a
a a a
a a a
a a a
-ª º°« »°« »°« »°« »
°« »°« »°« »®« »°« »°« »°« »°
« »°« »°« »
ai ∈ Z+ ∪ {0}, 1
Take P =
1 2 3
4 5 6
7 8 9
10 11 12
a a a
a a a
0 0 0
0 0 00 0 0
0 0 0
a a a
a a a
0 0 00 0 0
-ª º°« »°« »°« »°« »
°« »°« »°« »®« »°« »°« »°« »°
« »°« »°« »°¬ ¼¯
ai ∈ Z+ ∪ {0}, 1 ≤ i ≤
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 219/344
is an ideal of CS+ under natural product ×n.
Now
M =
1 2 3
4 5 6
7 8 9
0 0 0
a a a
a a a
0 0 0
0 0 0
0 0 0
a a a
0 0 00 0 0
-ª º°« »°« »°« »
°« »°« »°« »°
« »®« »°« »°« »°
« »°« »°
ai ∈ Z+ ∪ {0}, 1 ≤ i ≤
Take x =
1 2 3
4 5 6
7 8 9
0 0 0
0 0 0
0 0 0
a a aa a a
a a a
0 0 0
0 0 0
0 0 00 0 0
ª º« »« »« »« »
« »« »« »« »« »« »« »
« »« »« »¬ ¼
and y =
1 2
4 5
7 8
10 11
a a
a a
a a
0 00 0
0 0
a a
0 0
0 00 0
ª « « « «
« « « « « « «
« « « ¬
0 0 0ª º
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 220/344
we see x ×n y =
0 0 0
0 0 0
0 0 00 0 0
0 0 0
0 0 0
0 0 00 0 0
0 0 0
0 0 0
ª º« »« »« »« »« »« »« »« »« »
« »« »« »« »« »¬ ¼
. No element in CS+ h
Example 5.56: Let RS+
=
be the semigroup of super row vectors under the natuproduct ×n.
Take I =
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 11 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
ª º« »« »« »« »¬ ¼
be the unit i
Consider X =
1 5 9 13
2 6 10 14
3 7 11 15
4 8 12 16
a 0 0 a a a 0 0
a 0 0 a a a 0 0
a 0 0 a a a 0 0
a 0 0 a a a 0 0
-ª º
°« °« ®« °« °¬ ¼¯
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 221/344
¬ ¼¯
ai ∈ 5Z
+
∪ {0}, 1 ≤ i ≤ 16} ⊆ RS+
;
× is only a subsemigroup under ×n. X has no identiX is not an ideal of RS+ . However X has zero divisors
Take
Y =
1 2 9 10
3 4 11 12
5 6 13 14
7 8 15 16
0 a a 0 0 0 a a
0 a a 0 0 0 a a
0 a a 0 0 0 a a
0 a a 0 0 0 a a
-ª º°« »°« »®« »°« »°
¬ ¼¯
ai ∈ Q+
1 i 16} S+
Example 5.57: Let
4 4S+× =
1 2 3 4 5
6 7 8 9 10
11 12 13 14 15
16 17 18 19 20
21 22 23 24 25
a a a a a
a a a a a
a a a a aa a a a a
a a a a a
-ª º°« »°« »°« »®
« »°« »°« »°¬ ¼¯
ai ∈ Q+ ∪
be the semigroup of super square matrices unproduct ×n.
1 1 1 1 1
1 1 1 1 1
ª º« »
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 222/344
I =
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
« »« »« »
« »« »« »¬ ¼
acts as the identity with respect to the natural pro
Consider
P =
1 2 3
4 5
6 7
8 10 11
9 12 13
a 0 0 a a
0 a a 0 0
0 a a 0 0
a 0 0 a a
a 0 0 a a
-ª º°« »°« »°« »®
« »°« »°« »
°¬ ¼¯
ai ∈ Z+
Consider
M =
1 2
3 5 6
4 7 8
9 10 10 11
11 12 12 13
0 a a 0 0
a 0 0 a aa 0 0 a a
0 a a a a
0 a a a a
-ª º°
« »°« »°« »®« »°« »°« »°¬ ¼¯
ai ∈ Q+ ∪
1 ≤ i ≤ 12} ⊆ 4 4S
+
× ;
M is an ideal of 4 4S+
× . Every element p in P is such th
(0) for every m ∈ M.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 223/344
Inview of this we have the following theorem.
THEOREM 5.15: Let C S + (or RS + or m nS +
× (m ≠ n) o
semigroup under the natural product. Every ideal
RS+ or n mS+
× (m ≠ n) or n nS +
× ) is a subsemigroup of S
or n mS +
× (m ≠ n) or n nS+
× ) but however every subse
C S + (or RS + or
n mS +
× (m ≠ n) or n nS +
× ) need not in ge
ideal of C
S + (or R
S + or n m
S +
× (m ≠ n) or n m
S +
× ).
The proof is simple and direct hence left as an the reader.
and is not a semifield as it has zero divisors un×n.
Likewise RP+ = { RS+ , +, ×n} is a semiring
matrices which is not a semifield, infact a strisemiring.
m nP+
× (m ≠ n) = { m nS+
× (m ≠ n), +, ×n} is a str
semiring of m × n super matrices. Finally n nP+
×
is a strict commutative semiring of super square m
Now throughout this book CP+ will denote
super column matrices, RP+ will denote the sem
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 224/344
row matrices, m nP+
× (m ≠ n) will denote the sem
super matrices and n nP+
× will denote the semsuper matrices.
Now having seen the notation we procexamples of them.
Example 5.58: Let
1
2
3
4
a
a
a
a
-ª º°« »°« »°« »°
« »°« »°« »
be the semiring of super column matrices; CP+ is notas it has zero divisors.
Further CP+ has subsemirings for take
M =
1
2
3
4
a
a
a
a
0
0
0
-ª º°« »°« »°« »°« »°
« »°« »°« »®« »°« »°« »°
ai ∈ Z+ ∪ {0}, 1 ≤ i ≤ 6, +, ×n} ⊆
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 225/344
5
6
a
a0
« »°« »°« »
°« »°« »°¬ ¼¯
is a subsemiring of CP+ and is not an ideal of CP+ .
However CP+ has ideals for consider
0
0
0
0
-ª º°« »°« »°« »°« »
°« »°« »°
is an ideal of CP+ .
We see for every x ∈ M is such that
x ×n y =
00
0
0
0
0
0
0
ª º« »« »« »« »« »« »
« »« »« »« »« »« »
for every y ∈ N.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 226/344
0
0
« »« »
« »¬ ¼
Thus CP+ has infinitely many zero divisors so is n
Finally
1
11
1
1
1
1
ª º« »« »« »« »« »« »« »« »« »« »
acts as the identity with respe
has zero divisors for take x = (4 | 0 3 | 2 0 1 | 0 3 9 (0 | 7 0 | 0 8 0 | 9 0 0 | 1 0) in RP+ . Clearly x ×n y = (
0 | 0 0 0 | 0). So RP+ is a semiring which is not a sehas no identity.
Example 5.60: Let
3 4P+ =
1 12 23
2 13 24
3 14 25
4 15 26
5 16 27
6 17 28
a a a
a a a
a a aa a a
a a a
a a a
-ª º°« »°« »
°« »°« »°« »°« »°« »°®« » ai ∈ Q+ ∪ {0} 1 ≤ i ≤ 3
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 227/344
3 4P × 6 17 28
7 18 29
8 19 30
9 20 31
10 21 32
11 22 33
a a a
a a a
a a a
a a a
a a a
a a a
®« »°« »
°« »°« »°« »°« »°« »°« »
¬ ¼°̄
ai ∈ Q ∪ {0}, 1 ≤ i ≤ 3
be the semiring of super column vectors. 3 1P+
×
divisors, units, ideals and subsemirings which are However 3 11P+
× is not a semifield.
Example 5.61: Let 12 2P+
× =
semifield. Has ideals. Thus 12 2P+
× is a supsemiring.
Example 5.62: Let
4 4P+
× =
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
a a a a
a a a a
a a a a
a a a a
-ª º°« »°« »®« »°« »°« »¬ ¼¯
ai ∈ R+ ∪ {0}, 1
be the semiring of square super matrices. 4 4P+
× hunits, subsemirings which are not ideals and ideal
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 228/344
Clearly
1 1 1 1
1 1 1 11 1 1 1
1 1 1 1
ª º
« »« »« »« »« »¬ ¼
= I is the unit of 4 4P+
× .
S =
1
2
3
4 5 6 7
a 0 0 0a 0 0 0
a 0 0 0
a a a a
-ª º°« »°« »®« »°« »°« »¬ ¼¯
ai ∈ R+ ∪ {0}, 1 ≤
is a subsemiring which is also an ideal of 4 4
P+
×
.
L is a subsemiring of 4 4P+
× which is not an ideal of P
for every x ∈ S and y ∈ L we have x ×n y = 0 for eve
Example 5.63: Let
6 4
P+
×
=
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
17 18 19 20
21 22 23 24
a a a a
a a a a
a a a a
a a a aa a a a
a a a a
-ª º°« »°« »°« »°
« »®« »°« »°« »°« »°¬ ¼¯
ai ∈ Q+ ∪ {0}, 1 ≤ i
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 229/344
be a semiring of 6 × 4 super matrices.
I =
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
ª º« »« »« »« »
« »« »« »« »¬ ¼
is the unit of 6 4P+
× under natural
Further
a 0 0 a-ª º
Now
T =
9 10
11 12
1 6
2 7
3 8
4 5
0 a a 0
0 a a 0a 0 0 a
a 0 0 a
a 0 0 a
0 a a 0
-ª º°« »°« »°« »°
« »®« »°« »°« »°« »°¬ ¼¯
ai ∈ Q+ ∪
1 ≤ i ≤ 12, +, ×n} ⊆ 6 4P+
×
is a subsemiring as well as an ideal of 6 4P+
× . N
S h T h h
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 230/344
every x ∈ S we have every y ∈ T are such tha
Thus 6 4P+
× is only a semiring and not a semifield.
Now we proceed onto define semifields of sup
Let RJ+ = {(x1 x2 | x3 x4 x5 | … | xn) | xi ∈ R
1 ≤ i ≤ n} ∪ {(0 0 | 0 0 0 | … | 0)}, ×, +n} be super row matrices.
For RJ+ has no zero divisors with respect tostrict commutative semiring.
Now
CJ+ =
1
2
3
4
n 1
n
m
m
mm
m
m−
-ª º°« »°« »
°« »°« »°« »®« »°« »°« »°« »
°« »°« »¬ ¼¯
#
#
mi ∈ Z+ (or Q+ or R+), 1 ≤ i ≤ n} ∪
is the semifield of super column matrices.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 231/344
n mJ+× (m ≠ n) =
11 12 1m
21 22 2m
n1 n2 nm
a a ... a
a a ... a
a a ... a
-ª º°« »°« »®« »°« »°« »¬ ¼¯
# # #aij ∈ R+
(or Z+ or Q+), 1 ≤ i ≤ n; 1 ≤ j ≤ m} ∪
0 0 ... 00 0 ... 0
0 0 ... 0
ª « « « « « ¬
# # #
the semifield of n × m super matrices.
∪
0 0 0 ... 0
0 0 0 ... 0
0 0 0 ... 0
ª º« »« »« »« »
« »¬ ¼
# # # #, ×n, +}
is the semifield of square super matrices.
Now we just give some examples of them.
Example 5.64: Let
1
2
a
a
-ª º°« »°« »°« »
0
0
ª º« »« »« »
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 232/344
CJ+ =
3
4
5
6
7
8
9
a
aa
a
a
a
a
°« »°« »°« »°« »®
« »°« »°« »°« »°« »
°« »°¬ ¼¯
ai ∈ Z+; 1 ≤ i ≤ 9} ∪
0
00
0
0
0
0
« »
« »« »« »« »« »« »« »« »« »¬ ¼
be the semifield of super column matrices. Thisubsemifields.
Example 5.65: Let RJ+ = {(a1 a2 | a3 a4 a5 a6 a7 | a
Example 5.66: Let
5 5J+× =
1 2 3 4 5
6 7 8 9 10
11 12 13 14 15
16 17 18 19 20
21 22 23 24 25
a a a a a
a a a a a
a a a a aa a a a a
a a a a a
--ª º°°« »°°« »°°« »®®
« »°°« »°°« »°°¬ ¼¯¯
ai ∈ R+; 1 ≤
∪
0 0 0 0 00 0 0 0 0
0 0 0 0 0
0 0 0 0 0
ª º« »« »« »« »« »« »
, ×n, +}
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 233/344
0 0 0 0 0« »¬ ¼
is the semifield of super square matrices. This sesubsemifields.
Example 5.67: Let
8 4J+
×=
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
17 18 19 20
a a a a
a a a a
a a a a
a a a a
a a a a
--ª º°°« »°°« »°°« »°°« »°°« »
®®« »°°« »
ai ∈ Q+; 1 ≤ i
∪
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
ª º« »« »« »« »
« »« »« »« »« »« »« »¬ ¼
, ×n, +}
is a semifield of super 8 × 4 matrices. Thisubsemifields.
Example 5.68:
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 234/344
9 3J+× =
1 2 3 4 5 6 7
10 11 12 13 14 15 16
19 20 21 22 23 24 25
a a a a a a aa a a a a a a
a a a a a a a
--ª °°« ®®« °°« ¬ ¯¯
ai ∈ R+; 1 ≤ i ≤ 27}
∪
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
ª º« »« »« »¬ ¼
, +
is the semifield of super row vectors. This has su
Example 5.69: Let
J+ =
1 2
3 4
5 6
7 8
9 10
11 12
13 14
15 16
17 18
19 20
a a
a a
a aa a
a a
a a
a a
a a
a a
a a
--ª º°°« »°°« »
°°« »°°« »°°« »°°« »°°« »®®
« »°°« »°°« »°°« »°°« »°°« »°°« »°°¬ ¼¯¯
ai ∈ Z+; 1 ≤ i ≤ 20} ∪
0 0
0 0
0 00 0
0 0
0 0
0 0
0 0
0 0
0 0
ª º« « « « « « « « «
« « « « « ¬ ¼
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 235/344
be the semifield of super column vectors. Thsubsemifields but has subsemirings.
Chapter Six
SUPERMATRIX LINEAR ALGEBRA
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 236/344
In this chapter we introduce the notion o
vector space (linear algebra) and super matrix, posuper matrix coefficients. Several properties e
are defined, described and discussed.
Let V = {(x1 x2 | x3 … | xm-1 xn) | xi Q (or R
the collection of super row vectors of same type
V is a vector space over Q (or R). Now we
natural product on V so that V is a super linear a
row matrices or linear algebras of super row m
row matrices linear algebras.
For x = (x1 x2 x3 | x4 | x5 x6) and y = (y1 y2 y3 | y
have x un y = (x1y1 x2y2 x3y3 | x4y4 | x5y5 x6y6).
Example 6.2: Let
V = {(x1 x2 | x3 | x4 x5 x6 | x7 x8 | x9) | xi Q; 1 d
be a linear algebra of super row matrices over the fi
natural product un.
Consider P1
= {(x1
x2
| 0 | 0 0 0 | 0 0 | 0) | x1, x
2P2 = {(0 0 | a1 | 0 0 0 | a2 a3 | 0) | a1, a2, a3 Q} V a
0 | 0 | a1 a2 a3 | 0 0 | a4) | ai Q; 1 d i d 4} subalgebras of V over the field Q.
Clearly V = P1 + P2 + P3 and Pi Pj = (0 0 | 0 |
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 237/344
Clearly V = P1 + P2 + P3 and Pi P j = (0 0 | 0 |
0) if i z j; 1 d i, j d 3, so V is a direct sum of P1, P2 an
Example 6.3: Let
V = {(x1 | x2 x3 | x4 x5 | x6 x7 x8 x9 x10) | xi Q;
be a super row matrix linear algebra over the field Q
M1 = {(a1 | 0 | 0 | 0 0 | 0 0 0 a2 a3) | a1, a2, a3 Q}{(0 | a1 | 0 | 0 0 | a2 0 0 0 a3) | a1, a2, a3 Q} V, M
a1 | a2 0 | 0 0 0 0 a3) | a1, a2, a3 Q} V, M4 = {(0 |
0 a2 0 0 a3) | a1, a2, a3 Q} V and M5 = {(0 | 0 | 0
a2 a3) | a1, a2, a3 Q} V be a super sublinear alg
over Q.
Consider X = (x1 x2 x3 | 0 | 0 | 0 0 x4) where
4} V and Y = {(0 0 0 | x1 | x2 | x3 x4 0) | xi Q
be two linear subalgebras of super row matri
every x X and y Y, x un y = (0 0 0 | 0 | 0 | 0
see XA = Y and YA = X. Further V = X+Y and X
0 | 0 | 0 0 0).
We have seen examples of super linear al
row matrices.
Example 6.5: Let
1
2
a
a
-ª º°« »°« »°« »
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 238/344
V =
3
4
5
6
7
8
a
a
a
a
a
a
« »°« »°« »°« »®« »°« »°« »°« »°« »
°« »¬ ¼¯
ai Q; 1 d i d 8}
be a super column linear algebra over the field Q
V over Q is eight. Consider
0ª º« » xª º« »
in V is such that
x un y =
0
0
0
0
0
0
0
0
ª º« »« »« »
« »« »« »« »« »« »« »« »¬ ¼
.
Now we can find sublinear algebras of V.
Example 6.6: Let
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 239/344
V =
1
2
3
4
5
6
7
8
9
10
a
a
a
a
a
a
a
a
a
a
-ª º°« »°« »°« »°« »°« »°« »
°« »°« »®« »°« »°« »°« »°« »°
« »°« »°
ai Q; 1 d i d 11}
Consider
M1 =
1
2
a
a
0
0
0
0
0
00
0
0
-ª º°« »°« »°« »
°« »°« »°« »°« »°
« »®« »°« »°
« »°« »°« »°« »°« »°« »°¬ ¼¯
a1 , a2 Q} V,
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 240/344
M2 =
1
2
3
0
0
a
a
a0
0
0
0
0
-ª º°« »°« »°« »°« »°« »°« »°« »°
« »®« »°« »°« »°« »°« »°« »°« »°
a1, a2, a3 Q} V
M3 = 1
2
0
0
0
0
0
a
a
0
00
0
-ª º°« »°« »°« »°« »°
« »°« »°« »°
« »®« »°« »°« »°« »°« »°« »°« »°« »°¬ ¼¯
a1 , a2 Q} V
and
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 241/344
and
M4 =
1
2
3
0
0
0
0
00
0
a
a
a
-ª º°« »°« »°« »°« »°« »°« »°« »°
« »®« »°« »°« »°« »°« »°
« »°« »°
ai Q; 1 d i d 4} V
Clearly Mi M j =
0
0
0
0
0
0
0
0
00
0
ª º« »« »« »« »« »« »« »« »« »« »« »« »
« »« »« »« »¬ ¼
if i z j, 1 d i, j d 4.
Also V = M1 + M2 + M3 + M4; thus V is a dire
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 242/344
1 2 3 4;
subalgebras of V over Q.
We see for every x M1 every y M2 is su
(0). Likewise for every x M1, every z M3 is
= (0) and for every x M1, every t M4 is su
(0).
Hence we can say for every x M j every ele
j) (i=1 or 2 or 3 or 4) is orthogonal with x; how
Mi for i z j, i = 1 or 2 or 3 or 4.
Thus we see 1mA
= (M2 + M3 + M4); similar
Example 6.7: Let
P =
1
2
3
4
5
6
7
x
x
x
x
x
x
x
-ª º°« »°« »
°« »°« »°®« »°« »°« »°« »°« »
¬ ¼°̄
xi Q; 1 d i d 7}
be a super linear algebra of super column matrice
natural product un.
Consider
0-ª º
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 243/344
M1 =
1
2
3
0
x
x
0
0
0x
-ª º°« »°« »°« »°« »°®« »°« »°« »
°« »°« »¬ ¼°̄
xi Q; 1 d i d 7} P
and
1x
0
0
-ª º°« »
°« »°« »
We see M1 M2 =
0
0
0
0
0
0
0
ª º« »« »« »« »« »« »« »« »« »¬ ¼
and P = M1 +
Further the complementary subspace of M1 iversa. We see every element in M1 is orthogo
element in M2 under orthogonal product.
Example 6.8: Let
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 244/344
V =
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
a a a aa a a a
a a a a
a a a a
-ª º°« »°« »®« »°« »°« »¬ ¼¯
ai Q; 1 d
be a super linear algebra of square super matrice
product un.
We see for
1 2 30 a a aª º« »1x 0ª«
x un y =
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
ª º« »« »« »« »« »¬ ¼
in V
take y1 =
1 2 30 a a a
0 0 0 0
0 0 0 0
0 0 0 0
ª º« »« »« »« »
« »¬ ¼
in V
we see x un y1 =
0 0 0 0
0 0 0 0
0 0 0 0
ª º« »« »« » .
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 245/344
0 0 0 0« »« »¬ ¼
Thus we call y1 as a partial complement, only y is
total complement of x.
We can have more than one partial complement only one total complement.
Example 6.9: Let
1 2 3 4 5a a a a a
a a a a a
-ª º
°« »°« »
Consider
P1 =
1 3 4
2 5 6
a 0 0 a a
a 0 0 a a0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-ª º°« »°« »°« »®
« »°« »°« »°¬ ¼¯
ai Q; 1 d i d
P2 =
4
5
1 6
2 7
0 0 0 a 0
0 0 0 a 0
a 0 0 0 a
a 0 0 0 a
-ª º°« »°« »°« »®
« »°« »°« »
ai Q; 1 d
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 246/344
3 8a 0 0 0 a« »
°¬ ¼¯
P3 =
1 2 5
3 4 6
0 a a a 0
0 a a a 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-ª º°« »°« »°« »®
« »°« »°« »°¬ ¼¯
ai Q; 1 d
5
6
0 0 0 a 0
0 0 0 a 0
-ª º°« »
°« »°« »
P5 =
4
5
1
2
3
0 0 0 a 0
0 0 0 a 0
0 0 a 0 0
0 0 a 0 0
0 0 a 0 0
-ª º°« »°« »°« »®
« »°
« »°« »°¬ ¼¯
ai Q; 1 d i d 5}
are super linear subalgebras of super square matrices
We see V P1 + P2 + P3 + P4 + P5 and
Pi P j =
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
ª º« »« »« »« »« »
;
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 247/344
0 0 0 0 0
« »« »¬ ¼
if i z j; 1 d i, j d 5.
Thus V is only a pseudo direct sum of linear
and is not a direct sum of linear subalgebras.
Example 6.10: Let
V =
1 2 3
4 5 6
7 8 9
a a a
a a a
a a a
-ª º°« »°®« »
°« »¬ ¼°̄
ai Q; 1 d i d 9}
and
P2 =
1 2 3
4
5
a a a
0 0 a
0 0 a
-ª º°« »°®« »°« »
¬ ¼°̄
ai Q; 1 d i d 5}
be super linear subalgebras of V over Q. C
orthogonal with P1 is P2 and vice versa. No ocan be orthogonal (complement) of P1 in V.
Further V = P1 + P2 and P1 P2 =
0 0 0
0 0 0
0 0 0
ª
« « « ¬
Example 6.11: Let
a a a-
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 248/344
M =
1 2 3
4 5 6
7 8 9
10 11 12
13 14 15
16 17 18
19 20 21
a a a
a a a
a a a
a a a
a a a
a a a
a a a
-ª º°« »
°« »°« »°« »°®« »°« »°« »°« »°« »
¬ ¼°̄
ai Q; 1 d i d
be a super linear algebra of super column vecto
product un. Consider
P2 =
1 2 3
4 5 6
0 0 0
a a a
a a a
0 0 0
0 0 0
0 0 0
0 0 0
-ª º°« »°« »°« »
°« »°®« »°« »°« »°« »°« »
¬ ¼°̄
ai Q; 1 d i d 6} M
P3 =
0 0 0
0 0 0
0 0 0
0 0 0
-ª º°« »°« »°« »°« »°®« » ai Q; 1 d i d 6} M
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 249/344
1 2 3
4 5 6
0 0 0
a a a
a a a
°« »°« »°« »°« »
¬ ¼°̄and
P4 =1 2 3
4 5 6
0 0 0
0 0 0
0 0 0
a a a
a a a
0 0 0
-ª º°« »°« »°« »°« »°®« »°« »°« »°« »°
ai Q; 1 d i d 6}
We see P1 + P2 + P3 + P4 = V and Pi P j =
0 0
0 0
0 0
0 0
0 0
0 0
0 0
ª « « « « « « « « « ¬
1 d i, j d 4.
Example 6.12: Let
1 2 3 4
5 6 7 8
a a a a
a a a a
-ª º°« »°« »
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 250/344
M =
9 10 11 12
13 14 15 16
17 18 19 20
21 22 23 24
25 26 27 28
29 30 31 32
33 34 35 36
a a a a
a a a a
a a a a
a a a a
a a a a
a a a a
a a a a
« »°« »°« »°« »°« »®« »°« »°« »°
« »°« »°« »°¬ ¼¯
ai Q; 1 d
be a super linear algebra of super column vectors
P1 =
1 2 3 4
5 6 7 8
9 10 11 12
a a a a
0 0 0 0
a a a a
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
a a a a
-ª º°« »°« »°« »°« »°« »°« »®« »°« »°
« »°« »°« »°« »
°¬ ¼¯
ai Q; 1 d i d 12}
1 2 3 4
0 0 0 0
a a a a
0 0 0 0
-ª º°« »°« »°« »
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 251/344
P2 =
5 6 7 8
9 10 11 12
a a a a
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0a a a a
°« »°« »°« »®« »°« »°
« »°« »°
« »°« »°¬ ¼¯
ai Q; 1 d i d 12}
0 0 0 0
0 0 0 0
0 0 0 0
-ª º°« »°« »°« »°
P4 =
1 2 3 4
5 6 7 8
9 10 11 12
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
a a a a
a a a a
a a a a
-ª º°« »°« »°« »°« »°« »°« »®« »°« »°« »°« »°« »°« »°¬ ¼¯
ai Q; 1 d i d
be super linear subalgebras M over the field Q.
0 0 0 0
0 0 0 0
0 0 0 0
ª º« »« »« »
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 252/344
Clearly Pi P j z0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
« »« »« »« »« »« »« »« »« »¬ ¼
; if i z j; 1
Further V P1 + P2 + P3 + P4; thus V is
direct sum of super linear subalgebras of super
over Q.
be the super linear algebra of super row vectors over
Consider
P1 =
1 5
2 6
3 7
4 8
a a 0 0 0 0 0 0 0 0
a a 0 0 0 0 0 0 0 0
a a 0 0 0 0 0 0 0 0
a a 0 0 0 0 0 0 0 0
-ª °« °« ®« °« °¬ ¯
ai Q; 1 d i d 8} M,
P2 =
1 5
2 6
3 7
0 0 a a 0 0 0 0 0 0
0 0 a a 0 0 0 0 0 0
0 0 a a 0 0 0 0 0 0
-ª °« °« ®« °«
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 253/344
4 80 0 a a 0 0 0 0 0 0°¬ ¯
ai Q; 1 d i d 8} M,
P3 =
1 5
2 6
3 7
4 8
0 0 0 0 a a 0 0 0 0
0 0 0 0 a a 0 0 0 0
0 0 0 0 a a 0 0 0 0
0 0 0 0 a a 0 0 0 0
-ª
°« °« ®« °« °¬ ¯
ai Q; 1 d i d 8} M,
and
P5 =
1 2
4 5
7 8
10 11
0 0 0 0 0 0 0 0 a a
0 0 0 0 0 0 0 0 a a
0 0 0 0 0 0 0 0 a a0 0 0 0 0 0 0 0 a a
-ª °« °« ®
« °« °¬ ¯
1 d i d 12} M,
be super linear subalgebras of V of super row vec
Clearly Pi P j =
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
ª « « « «
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 254/344
0 0 0 0 0 0 0¬ i z j and 1 d i, j d 5.
M = P1 + P2 + P3 + P4 + P5 so M is the dire
linear subalgebras of super row vectors over the
Example 6.14: Let V =
1 2 3 4 5 6 7 8 9
11 12 13 14 15 16 17 18 19
a a a a a a a a a
a a a a a a a a a
-ª °®«
¬ °̄
1 d i d 20}
H2 =2 3
1 4
0 a a 0 0 0 0 0 0
0 a a 0 0 0 0 0 0
- ½ª º° °® ¾« »° °¬ ¼¯ ¿
H3 =
1 3
2 4
0 a 0 a 0 0 0 0 0
0 a 0 a 0 0 0 0 0
- ½ª º° °
® ¾« »° °¬ ¼¯ ¿
H4 =1 3
2 4
0 a 0 0 a 0 0 0 0
0 a 0 0 a 0 0 0 0
- ½ª º° °® ¾« »° °¬ ¼¯ ¿
H5 =1 3
2 4
0 a 0 0 0 a 0 0 0
0 a 0 0 0 a 0 0 0
- ½ª º° °® ¾« »° °¬ ¼¯ ¿
H6 =1 3
2 4
0 a 0 0 0 0 a 0 0
0 a 0 0 0 0 a 0 0
- ½ª º° °® ¾« »° °¬ ¼¯ ¿
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 255/344
H7 =1 3
2 4
0 a 0 0 0 0 0 a 0
0 a 0 0 0 0 0 a 0
- ½ª º° °® ¾« »° °¬ ¼¯ ¿
H8 = 1 3
2 4
0 a 0 0 0 0 0 0 a0 a 0 0 0 0 0 0 a
-ª º°®« »¬ ¼°̄
ai Q; 1 d
are super linear subalgebras of super row vect
field Q.
Clearly
Example 6.15: Let
V =
1 2 3 4 5
6 7 8 9 10
11 12 13 14 15
16 17 18 19 20
21 22 23 24 25
26 27 28 29 30
a a a a a
a a a a a
a a a a aa a a a a
a a a a a
a a a a a
-ª º°« »°« »
°« »°« »®« »°« »°« »°« »°¬ ¼¯
ai Q;
be a super linear algebra of 6 u 5 super matrices
under the natural product un.
Consider
1 0 0 0 0-ª º°
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 256/344
M =
1
2
3 4
a 0 0 0 0a 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 a a 0 0
-ª º°« »°« »°« »°
« »®« »°« »°
« »°« »°¬ ¼¯
ai Q; 1 d i
1
0 0 0 0 0
0 0 0 0 0
a 0 0 0 0
-ª º°« »°« »
°« »°« »® Q i
M3 =
1 2
3 4
0 a a 0 0
0 a a 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-ª º°« »°« »°« »°
« »®« »°« »°« »°« »°¬ ¼¯
ai Q; 1 d i d 4}
M4 =1 2
3 4
5 6
0 0 0 0 0
0 0 0 0 00 a a 0 0
0 a a 0 0
0 a a 0 0
0 0 0 0 0
-ª º°
« »°« »°« »°
« »®« »°« »°« »°« »°¬ ¼¯
ai Q; 1 d i d 4}
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 257/344
¬ ¼¯
M5 =
1 2
3 4
0 0 0 a a
0 0 0 a a
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
-ª º°« »°« »°« »°
« »®« »°« »°« »°« »°¬ ¼¯
ai Q; 1 d i d 4}
0 0 0 0 0
0 0 0 0 0
-ª º°« »°« »
M7 =1 2
3 4
5 6
0 0 0 0 0
0 0 0 0 0
0 0 0 a a
0 0 0 a a
0 0 0 a a
0 0 0 0 0
-ª º°« »°« »°« »°
« »®« »°« »°« »°« »°¬ ¼¯
ai Q; 1 d i
be super linear subalgebra of super matrices over
natural product un.
Clearly Mi M j =
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
ª º« »« »« »« »« »
« »« »
if i z j
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 258/344
0 0 0 0 00 0 0 0 0
« »« »« »¬ ¼
We see V = M1 + M2 + M3 + M4 + M5 + M6
is the direct sum of sublinear algebras of V.
Now we proceed onto give examtransformation and linear operators on super li
super matrices with natural product un.
Example 6.16: Let
1 2 3a a a-ª º°« »
Let
P =
1
2
3
4
5
6
7
8
9
a
a
a
a
a
a
a
a
a
-ª º°« »°« »°« »°
« »°« »°« »®« »°« »°« »°« »°« »°« »°¬ ¼¯
ai Q; 1 d i d 9}
be a super linear algebra of super column matrices ov
Q under the natural product un.
Define : M P by
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 259/344
Define K : M o P by
K1 2 3
4 5 6
7 8 9
a a a
a a a
a a a
§ ·ª º¨ ¸« »¨ ¸« »¨ ¸« »¬ ¼© ¹
=
1
2
3
4
5
6
7
8
9
a
a
a
a
a
a
a
a
a
ª º« »« »« »« »
« »« »« »« »« »« »« »« »¬ ¼
;
Example 6.17: Let
M =
1 2 3
4 5 6
7 8 9
10 11 12
13 14 15
16 17 18
a a a
a a a
a a aa a a
a a a
a a a
-ª º°« »°« »
°« »°« »®« »°« »°« »°« »°¬ ¼¯
ai Q; 1 d i d
be a super linear algebra of super matrices ounder the natural product un.
Let
P =1 3 5 6 7 11 13 15
2 4 8 9 10 12 14 16
a a a a a a a a
a a a a a a a a
-ª °®«
¬°̄
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 260/344
a a a a a a a a¬ °̄1 d i d 18}
be a super linear algebra of super matrices o
under natural product un.
Define K : M o P by
1 2 3
4 5 6
a a a
a a a
a a a
ª º« »« »« »
K is a linear transformation from M to P.
We now proceed onto define linear operator of
algebras of super matrices over the field F under natu
un.
Example 6.18: Let
V =
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
17 18 19 20
a a a a
a a a a
a a a aa a a a
a a a a
-ª º°« »°« »°
« »®« »°« »°« »°¬ ¼¯
ai Q; 1 d i d 2
be a super linear algebra of super matrices under
product un
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 261/344
product un.
Consider K : V o V defined by
K (
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
17 18 19 20
a a a a
a a a aa a a a
a a a a
a a a a
ª º« »« »« »« »« »« »¬ ¼
) =
1 2
3 4
5 6
7 8
0 a a 0
0 a a 00 a a 0
0 a a 0
0 0 0 0
§ ª ¨ « ¨ « ¨ « ¨ « ¨ « ¨ «
¬ ©
i il ifi d t b li t V
P is also a linear operator on V.
Example 6.19: Let
V =
1 2 3
4 5 6
7 8 9
10 11 12
13 14 15
16 17 18
19 20 21
a a a
a a a
a a a
a a a
a a a
a a aa a a
-ª º°« »°« »°« »°« »°®« »°« »°« »
°« »°« »¬ ¼°̄
ai Q; 1 d i d
be a super linear algebra of super column vecto
the field Q, under the natural product un.
fi
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 262/344
Define K : V o V
by K (
1 2 3
4 5 6
7 8 9
10 11 12
13 14 15
16 17 18
19 20 21
a a a
a a a
a a aa a a
a a a
a a a
a a a
ª º« »« »
« »« »« »« »« »« »« »¬ ¼
) =
1 2
4 5
7 8
10 11
a a
0 0
a a0 0
a a
0 0
a a
ª « «
« « « « « « « ¬
Example 6.21: Let
M =
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
17 18 19 20
21 22 23 24
25 26 27 28
a a a a
a a a a
a a a a
a a a a
a a a a
a a a a
a a a a
-ª º°« »
°« »°« »°« »°®« »°« »°« »°« »°
« »¬ ¼°̄
ai Q; 1 d i d
be a super linear algebra of super matrices over
under the natural product un.
Define f : M o Q by
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 263/344
o Q y
f (
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
17 18 19 20
21 22 23 24
25 26 27 28
a a a a
a a a a
a a a a
a a a aa a a a
a a a a
a a a a
ª º« »« »« »« »
« »« »« »« »« »¬ ¼
) = a1 + 3a3 + 9a2
f is a linear f nctional on M
from Q or Z or R}. S
R F [x] is defined as the su
coefficient polynomials or polynomials in the
row super matrix coefficients.
We will illustrate this situation by an exampl
Example 6.22: Let
S
R F [x] = i
ia x-®¯¦ ai (x1 | x2 x3 | x4 x5 x6
= {all super row matrices of the type (x1 | x2 xwith x j R, 1 d j d 7}} be the polynomials
matrix coefficients.
Example 6.23: Let
S
R F [x] =i
ia x
-
®̄¦ ai (x1 | x2 | x3 | x4)
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 264/344
[ ] ®̄¦ ( | | | )
{all super row matrices of the form (x1 | x2 | x3 |
1 d j d 4}} be the super row matrix coefficient po
We will illustrate how a polynomial looks and operations on them with super row matrix co
Let p(x) = (3 | 2 | –7 | 2 0) + (7 | –4 | 0 | 3 1
–7 –9)x5
be a super row matrix coefficient polyndegree of p(x) is 5.
Now we can add two polynomial with super row
and only if all the coefficients are from the same ty
row matrices.
Clearly we cannot add p(x) with q(x). However
can be added and q(x) + 0(x) = q(x).
We will illustrate addition of two super matrix po
Let m(x) = (1 1 | 0 2 3| 7 5 0 1) + (0 1 | 2 0 1 | 0 0
x | 1 0 0 | 8 0 0 5)x2
+ (0 0 | 0 0 1 | 20 1 2)x3
and n(x)
2 | 3 1 2 0) + (6 2 | 0 0 0 | 2 1 0 0)x + (0 1 | 1 0 1| 0 24 | 4 2 –1 | 0 7 2 1)x3 + (1 2 | 0 1 4 | 3 0 1 4)x4 be tw
matrix coefficient polynomials in the variable x.
m(x) + (n(x)) = (1 2 | 2 4 5 | 1 0 6 2 1) + (6 3 | 2
1)x + (0 9 | 2 0 1 | 8 2 0 6)x2 + (0 4 | 4 2 0 | 2 7 3 3)x3
4 | 3 0 1 4)x4
. Thus we see addition of two super coefficient polynomials is again a polynomial with
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 265/344
| ) pcoefficient polynomials is again a polynomial with
matrices coefficients. Infact the set of super
coefficient polynomials under addition is a group. Fu
commutative group under addition.
We will give examples of such groups.
Example 6.24: Let S
R F = i
i
i 0
a xf
-®¯¦ ai = (x1 | x2 | x3 x
{to the collection of all super row matrices of sam
t i f R} +} b b li f i fi it
Now we proceed onto give examples o
matrix coefficient polynomials.
p(x) =
3
2
0
1
1
4
5
ª º
« »« »« »« »« »« »« »
« »« »¬ ¼
+
0
1
2
0
1
4
0
ª º
« »« »« »« »« »« »« »
« »« »¬ ¼
x +
7
0
8
5
0
1
8
ª º
« »« »« »« »« »« »« »
« »« »¬ ¼
x3 +
8
0
7
0
0
1
9
ª º
« »« »« »« »« »« »« »
« »« »¬ ¼
x4 +
is a super column matrix coefficient polynomial
3
1
ª º« »« »
0
2
ª º« »« »
9
2
ª º« »« »
1
2
ª « «
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 266/344
Consider q(x) =
1
1
1
8
« »« »« »« »« »« »¬ ¼
+
2
3
4
0
« »« »« »« »« »« »¬ ¼
x +
2
3
1
8
« »« »« »« »« »« »¬ ¼
x2 +
2
3
4
0
«« « « « « ¬
is a column matrix coefficient polynomial of
variable x.
Now we show how addition of column m
polynomials are carried out in case of same
t i F if th l t i
Let p(x) =
3
2
1
0
1
5
ª º« »« »« »« »« »
« »« »« »¬ ¼
+
0
1
2
3
4
7
ª º« »« »« »« »« »
« »« »« »¬ ¼
x +
2
0
1
0
0
8
ª º« »« »« »« »« »
« »« »« »¬ ¼
x2 +
0
1
2
0
7
9
ª º« »« »« »« »« »
« »« »« »¬ ¼
x3 +
9
2
0
1
1
8
ª « « « « «
« « « ¬
q(x) =
9
0
1
2
5
0
ª º« »
« »« »« »« »« »« »« »¬ ¼
+
8
7
0
0
7
6
ª º« »
« »« »« »« »« »« »« »¬ ¼
x +
1
2
2
9
0
7
ª º« »
« »« »« »« »« »« »« »¬ ¼
x3 +
9
0
1
2
7
8
ª º« »
« »« »« »« »« »« »« »¬ ¼
x5.
ª º ª º § ª º ·ª º
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 267/344
p(x) + q(x) =
3
2
1
0
1
5
ª º« »« »« »« »« »« »« »« »¬ ¼
+
9
0
1
2
5
0
ª º« »« »« »« »« »« »« »« »¬ ¼
+
0
1
2
3
4
7
§ ª º¨ « »¨ « »¨ « »¨ « »¨ « »¨ « »¨ « »¨ « »¬ ¼©
+
8
7
0
0
7
6
·ª º¸« »¸« »¸« »¸« »¸« »¸« » ¸« »¸« »¬ ¼
x
2
0
§ ·ª º¨ ¸« »¨ ¸« »
0
1
§ ª º¨ « »¨ « »
1
2
·ª º¸« »¸« »
9
2
§ ª º¨ « »¨ « »
9
0
·ª º« »« »
=
12
2
2
2
6
5
ª º« »« »« »« »« »« »« »« »¬ ¼
+
8
8
2
3
12
13
ª º« »« »« »« »« »« »« »« »¬ ¼
x +
2
0
1
0
0
8
ª º« »« »« »« »« »« »« »« »¬ ¼
x2 +
1
3
4
9
7
16
ª º« »« »« »« »« »« »« »« »¬ ¼
x3 +
This is the way addition of super column ma polynomials are added. Thus addition is perfor
collection of all super column matrix coefficie
with same type of super column matrix coefficiengroup under addition.
We shall illustrate this situation by some sim
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 268/344
Example 6.26: Let S
CF [x] =
1
2
3i
4
i 0
5
6
7
a
a
aa x
a
a
a
f
- ª º° « »° « »
° « »° « »® « »° « »° « »° « »° « »
¬ ¼¯
¦ with
a
a
aa
a
a
a
ª « «
« « « « « « « ¬
Example 6.27: Let S
CF [x] = i
i
i 0
a xf
-®¯¦ ai =
1
2
3
4
5
6
x
x
x
x
x
x
ª º« »« »« »« »« »
« »« »« »¬ ¼
M
with x j Z, 1 d j d 6}, +} be an abelian group under
Example 6.28: Let
S
3 5F u = i
i
i 0
a xf
-®¯¦ ai =
1 2 3 4 5
6 7 8 9 10
11 12 13 14 15
d d d d d
d d d d d
d d d d d
ª º« »« »« »¬ ¼
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 269/344
{all 3 u 5 matrices with entries from Z, d j Z; 1 dan abelian group under addition.
Example 6.29: Let S
4 4F u = i
i
i 0
a xf
-®¯¦ ai =
1 2
5 6
9 10
13 14
x x
x xx x
x x
ª « « « « ¬
with x j Q; 1 d j d 16} be an abelian group und
Cl l SF h b
is a subgroup of G under addition.
Now we proceed onto give the semigroup
the natural product un.
Let S
CF = i
i
i 0
a xf
-®¯¦ ai =
1
2
20
xx
x
ª º« »« »« »« »« »¬ ¼
with x j Z,
be the collection of super column matrix coefficiS
CF is a semigroup under the natural product un.
commutative semigroup.
Suppose
1aª º 1 bª º 1cª º
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 270/344
p(x) =
1
2
20
a
a
ª º« »« »« »« »« »¬ ¼
+
1
2
20
b
b
ª º« »« »« »« »« »¬ ¼
x +
1
2
20
c
c
ª º« »« »« »« »« »¬ ¼
x3 a
q(x) =
1
2
20
d
d
d
ª º« »« »« »« »
« »¬ ¼
+
1
2
20
e
e
e
ª º« »« »« »« »
« »¬ ¼
x
2+
1
2
20
m
m
m
ª º« »« »« »« »
« »¬ ¼
x
p(x) un q(x) =
1 1
2 2
20 20
a d
a d
a d
ª º« »« »« »« »« »¬ ¼
+
1 1
2 2
20 20
b d
b d
b d
ª º« »« »« »« »« »¬ ¼
x +
1 1
2 2
20 20
a e
a e
a e
ª º« »« »« »« »« »¬ ¼
x2 +
+
1 1
2 2
20 20
c d
c d
c d
ª º« »« »« »« »
« »¬ ¼
x3 +
1 1
2 2
20 20
a m
a m
a m
ª º« »« »« »« »
« »¬ ¼
x4 +
1 1
2 2
20 20
c e
c e
c e
ª º« »« »« »« »
« »¬ ¼
x5 +
ª««««
«¬
+
1 1
2 2
20 20
b m
b m
b m
ª º« »« »« »« »« »¬ ¼
x5
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 271/344
=
1 1
2 2
20 20
a d
a d
a d
ª º« »« »« »
« »« »¬ ¼
+
1 1
2 2
20 20
b d
b d
b d
ª º« »« »« »
« »« »¬ ¼
x +
1 1
2 2
20 20
a e
a e
a e
ª º« »« »« »
« »« »¬ ¼
x2 +
1 1
2 2
m m
b e
b e
b e
ª « « «
« « ¬
+
1 1
2 2
a m
a m
ª º« »« »
« »« »
x4 +
1 1 1 1
2 2 2 2
c e b m
c e b m
ª º« »« »
« »« »
x5 +
1 1
2 2
c m
c m
ª º« »« »
« »« »
x7
Example 6.30: Let
S
3 6F u = i
i
i 0
a xf
-®¯¦ ai =
1 2 3 4 5
7 8 9 10 11
13 14 15 16 17
x x x x x
x x x x x
x x x x x
ª « « « ¬ 1 d j d 18}
be a super 3 u 6 matrix coefficient polynomial s
the natural product un.
Example 6.31: Let
S
3 3F u = i
i
i 0
a xf
-®¯¦ ai =
1 2 3
4 5 6
7 8 9
x x x
x x x
x x x
ª º« »« »« »¬ ¼
where x j =
be a suepr square matrix coefficient semigrou
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 272/344
be a suepr square matrix coefficient semigrou
product un.Let
p(x) =
3 2 0
1 0 1
0 2 3
§ ·¨ ¸¨ ¸¨ ¸© ¹
+
7 5 1
0 1 2
0 0 3
§ ·¨ ¸¨ ¸¨ ¸© ¹
x +
1
0
0
§ ¨ ¨ ¨ ©
+
0 0 9
1 0 3
§ ·¨ ¸
¨ ¸¨ ¸x4
be in S
3 3F u .
To find
p(x) un q(x); p(x) un q(x) =
3 2 0
1 0 10 2 3
§ ·¨ ¸
¨ ¸¨ ¸© ¹
un
4
17
§ ¨
¨ ¨ ©
+
3 2 0
1 0 1
0 2 3
§ ·¨ ¸¨ ¸¨ ¸© ¹
un
1 2 3
4 5 6
7 8 9
§ ·¨ ¸¨ ¸¨ ¸© ¹
x2 +
3 2 0
1 0 1
0 2 3
§ ·¨ ¸¨ ¸¨ ¸© ¹
un
0
2
3
§ ¨ ¨ ¨ ©
+
7 5 1
0 1 2
0 0 3
§ ·¨ ¸¨ ¸
¨ ¸© ¹
un
4 0 2
1 5 6
7 0 2
§ ·¨ ¸¨ ¸
¨ ¸© ¹
x +
7 5 1
0 1 2
0 0 3
§ ·¨ ¸¨ ¸
¨ ¸© ¹
un
1
4
7
§ ¨ ¨
¨ ©
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 273/344
+
7 5 1
0 1 2
0 0 3
§ ·¨ ¸¨ ¸¨ ¸
© ¹
un
0 3 1
2 1 0
3 4 5
§ ·¨ ¸¨ ¸¨ ¸
© ¹
x4 +
1 2 3
0 0 7
0 1 2
§ ·¨ ¸¨ ¸¨ ¸
© ¹
un
4
1
7
§ ¨ ¨ ¨
©
+
1 2 3
0 0 7
0 1 2
§ ·¨ ¸¨ ¸¨ ¸© ¹
un
1 2 3
4 5 6
7 8 9
§ ·¨ ¸¨ ¸¨ ¸© ¹
x4 +
1 2 3
0 0 7
0 1 2
§ ·¨ ¸¨ ¸¨ ¸© ¹
un
0
2
3
§ ¨ ¨ ¨ ©
=
12 0 0
1 0 6
0 0 6
§ ·¨ ¸¨ ¸¨ ¸© ¹
+
3 4 0
4 0 6
0 16 27
§ ·¨ ¸¨ ¸¨ ¸© ¹
x2 +
0 6
2 0
0 8
§ ¨ ¨ ¨ ©
+
28 0 2
0 5 12
0 0 6
§ ·¨ ¸¨ ¸¨ ¸© ¹
x +
7 10 3
0 5 12
0 0 27
§ ·¨ ¸¨ ¸¨ ¸© ¹
x3 +
0
0
0
§ ¨ ¨ ¨ ©
+
1 4 9
0 0 42
0 8 18
§ ·¨ ¸¨ ¸¨ ¸© ¹
x4 +
0 6 3
0 0 0
0 4 10
§ ·¨ ¸¨ ¸¨ ¸© ¹
x5 +
0
1
14
§ ¨ ¨ ¨ ©
+
0 0 27
4 0 18
§ ·¨ ¸¨ ¸
6 +
0 0 9
2 0 0
§ ·¨ ¸¨ ¸
7 +
4
0
§ ¨ ¨
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 274/344
+ 4 0 18
14 56 18
¨ ¸¨ ¸© ¹
x6 + 2 0 0
6 28 10
¨ ¸¨ ¸© ¹
x7 + 0
0
¨ ¨ ©
=
12 0 0
1 0 6
0 0 6
§ ·¨ ¸¨ ¸¨ ¸© ¹
+
28 0 2
0 5 12
0 0 6
§ ·¨ ¸¨ ¸¨ ¸© ¹
x +
7 4
4 0
0 16
§ ¨ ¨ ¨ ©
7 16 3§ · 1 19 28§ · 0§
Thus we see S
3 3F u [x] under natural produc
semigroup. This semigroup has zero divisors and
can derive all related properties of this semigroup as
routine.
Now we can also give these super matrix
polynomials a ring structure. We just recall if S
CF [x
column matrix polynomials, we know S
CF [x] under
an abelian group and under the natural product un,
semigroup. Thus it is easily verified ( SCF [x], +
commutative ring known as the super column matrix
ring.
We will illustrate this situation by some simple e
Example 6.32: Let
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 275/344
S
CF [x] =i
i
i 0a x
f
-
®̄¦ ai =
1
2
3
4
5
6
x
x
x
x
x
x
ª º« »« »« »
« »« »« »« »« »¬ ¼
with x j Z, 1 d j
be the super column matrix coefficient polynomial r
Example 6.33: Let ( S
CF [x], +, un) =i
i
i 0
a xf
-®¯¦
d j Q; 1 d j d 8, +, un} be the super column m
polynomial ring of infinite order. S
CF [x] has sub
not ideals, has zero divisors and idempotents
special form which are only constant polynomial
10
ª º« »« »
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 276/344
For instance D =
0
1
1
1
0
1
« »« »« »« »« »
« »« »« »« »« »¬ ¼
S
CF [x] is such tha
0ª º« »
Further we see P = i
i
i 0
a xf
-®¯¦
ai =
1
2
3
4
5
6
7
8
x
x
x
x
xx
x
x
ª º« »« »« »« »« »
« »« »« »« »« »« »¬ ¼
with x j
1 d j d 8, +, un} SCF [x] is a subring of super col
coefficient polynomial ring. However P is not an
[x].
However S
CF [x] has infinite number of zero divi
dª º
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 277/344
Example 6.34: Let S
CF [x] = i
i
i 0
a xf
-®¯¦ ai =
1
2
3
d
d
d
ª º« »« »« »¬ ¼
w
un, +, 1 d j d 3} be a super column matrix polynomial ring.
Consider p(x) = 1
0
a
ª º« »« »« »
+ 1
0
b
ª º« »« »« »
x + 1
0
c
ª º« »« »« »
x2
Clearly p(x) un q(x) =
0
0
0
ª º« »« »« »¬ ¼
.
Further if
P = i
i
i 0
a xf
-®¯¦ ai =
1
2
x
0
x
ª º« »« »« »¬ ¼
with x1, x2 Z, +, u
is a subring.
Also T = i
i
i 0
a xf
-®¯¦ ai = 1
0
y
0
ª º« »« »« »¬ ¼
, y1 Z, +, un
is a subring.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 278/344
We see S
CF [x] =
P + T and P T =00
0
ª º« »« »« »¬ ¼
.
Further for every D P we have every E T
coefficient polynomial ring which is commutati
infinite order.
We will give examples of it.
Example 6.35: Let R = { SR F [x] = i
i
i 0
a xf
¦ ; ai = (t1 t2
t j Z, 1 d j d 6, +, un} be a super row matrix
coefficient ring. R has zero divisors, units, ideals an
P =i
i
i 0a x
f
-®̄¦ ai = (0 0 0 | d1 | d2 d3) d1, d2, d3 Z, +
is a subring as well as an ideal of R.
T = i
i
i 0
a xf
-®
¯
¦ ai = (y1 y2 y3 | 0 | 0 0) y1, y2, y3 Z,
is a subring as well as an ideal of R.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 279/344
We see every a P and every b T are such th
(0 0 0 | 0 | 0 0).
Also (1 –1 1 | –1 | –1 1) = p is such that p2
= (1 1only units of this form are in R.
Example 6.36: Let S
R F [x] = i
i
i 0
a xf
-®¯¦ ai = (p1 p2 | p
T = i
i
i 0
a xf
-®¯¦ ai = (y1 y2 | y3 | y4 y5 | y6 | y7 y8 y9
d j d 9, +, un} S
R F [x]; T is a only a subring andS
R F [x].
It is easily verified S
R F [x] has zero divisors.
Example 6.37: Let S
R F [x] = i
i
i 0
a xf
-®¯¦ ai = (m1
R, +, un} be a super row matrix coefficient p
Clearly S
R F [x] has units, zero divisors, idempot
are only constant polynomials. For D = (1 | –1)
(1 | 1) and E = (0 | 1) and b1 = (1 | 0) are all id
also see P =
i
ii 0 a x
f
-
®̄¦ ai = (t | s), t, s, Z, +, u
subring and not an ideal of SF [x]
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 280/344
subring and not an ideal of R F [x].
Take M = i
i
i 0
a xf
-®
¯
¦ ai = (t | 0); t R, +, un
ideal of S
R F [x]. N = i
i
i 0
a xf
-®¯¦ ai = (0 | s) s
S
R R [x] is also an ideal of S
R F [x]. Every D in M
M are such that DunE = (0 | 0).
We will illustrate this by some examples.
Example 6.38: Let { S
2 4F u [x], +, un} =i 0
af
-®¯¦
1 2 3 4
5 6 7 8
d d d d
d d d d
ª º« »¬ ¼
d j z, 1 d j d 8, +, un} be the
vector coefficient polynomial ring. S
2 4F u has zero div
idempotents ideals and subrings.
D =1 1 1 1
1 1 1 1
ª º« » ¬ ¼
is such that D2 =1 1
1 1
ª « ¬
unit.
Consider E =1 0 1 1
0 1 0 0
ª º« »¬ ¼
inS
2 4F u [x]. We se
E i id t t i SF [ ] l if it i t t
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 281/344
E is an idempotent in S
2 4F u [x] only if it is a constant
and the super row vector takes its entries only as 0 or
Similarly an element a S
2 4
Fu
[x] is a unit on
constant polynomial and all its entries are from the
Consider
P = i
i
i 0
a xf
-®¯¦ ai =
1 2 3 4
5 6 7 8
t t t t
t t t t
ª º« »¬ ¼
; t j
Example 6.39: Let
{ S
5 3F u [x], +, un} = i
i
i 0
a xf
-®¯¦ ai =
1 2 3
4 5 6
7 8 9
10 11 12
13 14 15
y y y
y y y
y y y
y y yy y y
ª « « « «
« « ¬
1 d j d 15, un, +}
be the super column vector coefficient polynomihas subrings which are not ideals, ideals, unit
and idempotents.
Consider M = i
i
i 0
a xf
-®¯¦ ai =
1 2
4 5
7 8
m m
m m
m m
m m m
ª «
« « « «
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 282/344
10 11
13 14
m m m
m m m
« « ¬
z, 1 d j d 15, +, un} S
5 3F u [x]
is only a subring and not an ideal of S5 3F u [x].
Take p =
1 0 1
1 1 1
0 0 1
ª º« »« »« » in S
5 3F u [x] is such that p
Consider t =
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
ª º« »« »« »« »
« »
« » ¬ ¼
in S
5 3F u [x].
We see t2 =
1 1 1
1 1 1
1 1 1
1 1 1
1 1 1
ª º« »« »« »« »
« »« »¬ ¼
the unit; that t is a un
Suppose y S
5 3F u [x] is to be a unit then we see
a constant polynomial and the 5 u 3 matrix must tak
from the set {1, –1}.
Example 6.40: Let
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 283/344
p
S
5 4F u [x] = i
i
i 0
a xf
-®¯¦ ai =
1 6 11
2 7 12
3 8 13
4 9 14
5 10 15
m m m m
m m m m
m m m m
m m m m
m m m m
ª «
« « « « « ¬
Take M = i
i
i 0
a xf
-®¯¦ ai =
1 6 11
2 7 12
3 8 13
4 9 14
5 10 15
m m m
m m m
m m m
m m m
m m m
ª « « « « «
« ¬
with m j 5Z, 1 d j d 20, +, un} S
5 4F u [x];
M is an ideal of S
5 4F u [x].
Suppose
P = i
i
i 0
a xf
-®¯¦ ai =
1
3 4
5 6
7 8
2
m 0 0 m
0 m m
0 m m
0 m m
m 0 0 m
ª « « « « « «
¬
with m 3Z 1 d j d 10 + u } SF
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 284/344
with m j 3Z, 1 d j d 10, +, un} 5F u
be an ideal of S
5 4F u [x].
Take T = i
i
i 0
a xf
-®¯¦ ai =
4 5
1 7
2 8
3 9
0 m m 0
m 0 0 m
m 0 0 m
m 0 0 m
ª « « « « «
T D un E =
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
ª º« »« »« »« »« »
« »¬ ¼
.
However P T =
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
ª º« »« »« »
« »« »« »¬ ¼
but P T z S
3F
Example 6.41: Let
S5 7F u [x] = i
i
i 0
a x
f
-®¯¦ ai =
m m m m m m mª º
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 285/344
1 2 3 4 5 6 7
8 9 10 11 12 13 14
15 16 17 18 19 20 21
22 23 24 25 26 27 28
29 30 31 32 33 34 35
m m m m m m m
m m m m m m m
m m m m m m m
m m m m m m m
m m m m m m m
ª º« »« »« »
« »« »« »¬ ¼
with
1 d j d 35, +, un}
S
m =
1 1 1 1 1 1 1
1 1 1 1 1 1 1
1 1 1 1 1 1 1
1 1 1 1 1 1 1
1 1 1 1 1 1 1
ª º« » « »« » « »
« »
« » ¬ ¼
in S
5F u
is such that
m2 =
1 1 1 1 1 1 1
1 1 1 1 1 1 11 1 1 1 1 1 1
1 1 1 1 1 1 1
1 1 1 1 1 1 1
ª º« »
« »« »« »« »« »¬ ¼
S
5 7F u [
F h if
1 0 0 1 0 1 10 1 1 0 1 0 0
0 1 1 0 0 1 1
ª º« »« »« » SF
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 286/344
Further if p = 0 1 1 0 0 1 1
1 0 1 0 1 0 1
1 1 1 1 0 1 0
« »« »« »« »¬ ¼
S
5F u
then p2
= p is an idempotent of S
5 7F u [x].
Take
P = i
i
i 0
a xf
-®¯¦ ai =
1 6 11
2 7 12
3 8 13
4 9 14
5 10 15
m 0 0 m 0 m
m 0 0 m 0 m
m 0 0 m 0 m
m 0 0 m 0 mm 0 0 m 0 m
ª « « « «
« « ¬
m j Z; 1 d j d 15, +, un}
to be a subring and not an ideal of S
5 7F u [x].
Take
S = i
i
i 0
a xf
-®¯¦
ai =
1 2 11
3 4 12
5 6 13
7 8 14
9 10 15
0 m m 0 m 0
0 m m 0 m 0
0 m m 0 m 0
0 m m 0 m 0
0 m m 0 m 0
ª « « «
« « « ¬
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 287/344
9 10 15¬
with m j Q; 1 d j d 20, +, un} is an ideal of S
5F
We see every polynomial p(x) P is such that for ev
0 0 0 0 0
0 0 0 0 0
ª « «
Let S
m mF u [x] be the collection of a super
coefficient polynomial ring under + and un. W
this by some examples.
Example 6.42: Let
S
4 4F u [x] = i
i
i 0
a xf
-®¯¦ ai =
1 2 3
5 6 7
9 10 11
13 14 15
m m m
m m m
m m m
m m m
ª « « « « ¬
m j Z, 1 d j d 16, +, un}
be the super square matrix coefficient polynomia
has zero divisors units, idempotents, ideals and s
Take D =
1 1 1 11 1 1 0
1 0 1 0
ª º« »« »« »
S
4 4F u [x], we se
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 288/344
1 0 1 0
0 1 1 1
« »« »¬ ¼
is not a unit.
Take E =
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
ª º« » « »« »« »
¬ ¼
S
4 4F u [x];
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 289/344
1. Every constant polynomial with entr
{1, –1} is a unit under u n.2. Every constant polynomial with entr
{0, 1} is an idempotent under u n.
THEOREM 6.2:
Every super matrix coefficient phas ideals.
THEOREM 6.3: Every super matrix coefficient pover Q or R has subrings which are not ideals.
THEOREM 6.4:
Every super matrix coefficient phas infinite number of zero divisors.
Now we can define super vector space of po
R or Q.
Suppose we take V =
S
CF [x] to be an abeliaddition. V is a vector space over the reals or rat
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 290/344
We will give examples of them.
Example 6.43: Let
V = { S
CF [x] = i
i
i 0
a xf
¦ with ai =
1
2
3
4
5
t
t
t
t
t
ª º« »« »« »« »« »« »
; t j Q
Example 6.44: Let
V = { S
CF [x] = i
i
i 0
a xf
¦ with ai =
1
2
3
t
t
t
ª º« »« »« »¬ ¼
; t j Q, 1 d j
be an abelian group under ‘+’. V is a super col
coefficient polynomial vector space over Q.
P1 =i
i
i 0a x
f
-
®̄¦ ai =
1t
00
ª º« »« »« »¬ ¼
; t1 Q, +} V
is a subspace of V over Q.
P2 =i
i
i 0a x
f
-
®̄¦ ai = 2
0
t0
ª º« »« »« »¬ ¼
; t2 Q, +} V
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 291/344
is a subspace of V over Q.
P3 =i
i
i 0a x
f
-®̄¦ ai =
3
0
0t
ª º« »« »« »¬ ¼
; t3 Q, +} V
is a subspace of V over Q.
We see V = P1 + P2 + P3 and
Example 6.45: Let
M = i
i
i 0
a xf
-®¯¦ ai =
1
2
3
4
5
6
7
m
m
m
m
m
m
m
ª º« »« »« »« »« »« »« »« »« »
¬ ¼
; m j Q; 1 d j
be a super column matrix coefficient polynom
‘+’.
M is a super column matrix coefficient vecto
Take
1
2
m
m
0
ª º« »« »« »
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 292/344
P1 = i
i
i 0
a xf
-®¯¦ ai =
3
0
0
0
0
m
« »« »« »« »
« »« »« »¬ ¼
, m1, m2, m3 Q, +
1mª º« »
P3 = i
i
i 0
a xf
-®¯¦ ai =
1
2
3
4
m
0
0
0
mm
m
ª º« »« »« »« »« »
« »« »« »« »¬ ¼
with m j Q, 1 d j d 4
Clearly M P1 + P2 + P3 but Pi P j z
0
0
0
0
0
00
ª º
« »« »« »« »« »« »« »
« »« »¬ ¼
if i z
1 d i, j d 3. P1, P2 and P3 are subspaces of M over Q
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 293/344
, j 1, 2 3 p Q
is the pseudo direct sum of vector subspaces of M.
Example 6.46: Let
V = i
i
i 0
a xf
-®¯¦ ai = (d1 | d2 | d3 d4 d5 d6 d7 | d8); d
1 d j d 8}
P2 = i
i
i 0
a xf
-®¯¦ ai = (d1 | d2 | 0, 0, 0, 0, 0 | 0), d1,
P3 = i
i
i 0
a xf
-®
¯
¦ ai = (0 | 0 | 0 0 d1 d2 0 | 0); d1,
and
P4 = i
i
i 0
a xf
-®
¯
¦ ai = (0 | 0 | 0 … 0 d1 | d2) with d
be a super row matrix coefficient polynomials
of V over Q.
Clearly V = P1 + P2 + P3 + P4 and
Pi P j = ( 0 | 0 | 0 0 0 0 0 | 0) if i z j, 1 d
Cl l V i di t f b
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 294/344
Clearly V is a direct sum of subspaces.
Example 6.47: Let
V = i
i
i 0
a xf
-®¯¦ ai = (t1 t2 | t3 t4 | t5); t j Q,
be the super row matrix coefficient polynomi
M2 = i
i
i 0
a xf
-®¯¦ ai = (t1, t2 | 0 0 | 0), t1, t2 Q
M3 = i
i
i 0
a xf
-®
¯
¦ ai = (0 t1 | 0 t2 | 0), t1, t2 Q
and
M4 = i
i
i 0
a xf
-®¯¦ ai = (0 t1 | 0 0 | t2), t1, t2 Q}
be super row matrix coefficient polynomial vector
V over Q. We see V M1 + M2 + M3 + M4, howeveMi M j z (0 0 | 0 0 | 0) if i z j; 1 d i, j d 4
Thus V is only a pseudo direct sum of vector
over Q.
Example 6.48: Let
V = ia xf-
®¦ ai = (m | m m m | m m | m
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 295/344
V = i
i 0
a x
®¯¦ ai = (m1 | m2 m3 m4 | m5 m6 | m
with m j R; 1 d j d 7}
be a super row matrix coefficient polynomial vector
Q.
Consider
to be super row matrix coefficient polynomial v
of V over Q.
We see V = M1 + M2 and M1 M2 = ( 0 | 0 0
Infact we see every q(x) M1 is orthogo
other p(x) M2.
Thus M1 is the orthogonal complement o
versa. Now we give examples of super m u n m
polynomial vector spaces over Q or R.
Example 6.49: Let
- 1 4 7 10 13 16d d d d d dª
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 296/344
V = i
i
i 0
a xf
-®¯¦ ai =
1 4 7 10 13 16
2 5 8 11 14 17
3 6 9 12 15 18
d d d d d d
d d d d d d
d d d d d d
ª « « «
¬ 1 d j d 18}
be the super 3 u 7 row vector coefficient polspace over Q.
Example 6.50: Let
W = i
i
i 0
a xf
-®¯¦ ai =
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
17 18 19 20
21 22 23 24
25 26 27 28
29 30 31 32
33 34 35 36
37 38 39 40
m m m m
m m m m
m m m m
m m m m
m m m m
m m m m
m m m m
m m m m
m m m m
m m m m
ª « «
« « « « « « «
« « « « « ¬
with m j R, 1 d j d 40}
be a super column vector coefficient polynomial v
over Q.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 297/344
Q
Example 6.51: Let
V = i
i
i 0
a xf
-®¯¦ ai =
1 2 3 4
6 7 8 9
11 12 13 14
16 17 18 19
21 22 23 24
m m m mm m m m m
m m m m m
m m m m m
m m m m m
ª « « « « « «
Now we see for all these one can easisubspaces etc.
Example 6.52: Let
V = ii
i 0
a x
f
-®¯¦ ai =
1 8 15 22 29 36 43
2 9 16 23 30 37 44
3 10 17 24 31 38 45
4 11 18 25 32 39 46
5 12 19 26 33 40 47
6 13 20 27 34 41 48
7 14 21 28 35 42 49
m m m m m m m
m m m m m m m
m m m m m m mm m m m m m m
m m m m m m m
m m m m m m m
m m m m m m m
ª « «
« « « « « « «
¬ m j Q, 1 d j d 56}
be a super 7 u 8 matrix coefficient polynomial ve
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 298/344
p p y
Q. Consider
M1 = i
i
i 0
a xf
-®¯¦ ai =
1
2
3
4
5
m 0 0 0 0 0m 0 0 0 0 0
m 0 0 0 0 0
m 0 0 0 0 0
m 0 0 0 0 0
ª « « « « « «
M2 = i
i
i 0
a xf
-®¯¦ ai =
1 2
3 4
5 6
7 8
9 10
11 12
13 14
0 m m 0 0 0
0 m m 0 0 0
0 m m 0 0 0
0 m m 0 0 0
0 m m 0 0 00 m m 0 0 0
0 m m 0 0 0
ª « « « « « « « « « ¬
with m j Q, 1 d j d 14} V,
M3 = i
i
i 0
a xf
-®
¯
¦ ai =
1 2 3
4 5 6
7 8 9
10 11 12
13 14 15
16 17 18
19 20 21
0 0 0 m m m
0 0 0 m m m
0 0 0 m m m
0 0 0 m m m
0 0 0 m m m
0 0 0 m m m
0 0 0 m m m
ª « « « « «
« « « « ¬
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 299/344
19 20 21¬
with m j Q, 1 d j d 21} V
and
if-
1
3
5
0 0 0 0 0 0 m
0 0 0 0 0 0 m
0 0 0 0 0 0 m
0 0 0 0 0 0 m
ª « « « «
Mi M j =
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
ª º« »« »« »« »
« »« »« »« »« »¬ ¼
,
Thus V is the direct sum of subspaces.
Example 6.53: Let
P =i
i
i 0a x
f
-
®̄¦ ai =
1 2 3 4 5
8 9 10 11 1
15 16 17 18 1
22 23 24 25 2
m m m m m
m m m m m
m m m m m
m m m m m
ª «
« « « ¬
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 300/344
with mi R, 1 d j d 28}
be a 4 u 7 super row vector matrix coefficient pospace over Q.
Let B1 = i
i
i 0
a xf
-®¯¦ ai =
1 5
2 6
3 7
m m 0 0
m m 0 0
m m 0 0
ª « « «
B2 = i
i
i 0
a xf
-®¯¦ ai =
1 5
2 6
3 7
4 8
m 0 m 0 0
m 0 m 0 0
m 0 m 0 0
m 0 m 0 0
ª « « « « ¬
mi Q, 1 d i d 8} P,
B3 = i
i
i 0
a xf
-®
¯
¦ ai =
1 5
2 6
3 7
4 8
m 0 0 m 0
m 0 0 m 0
m 0 0 m 0
m 0 0 m 0
ª « « « « ¬
mi Q, 1 d i d 8} P,
B4 = i
i
i 0
a xf
-®¯¦ ai =
1 5
2 6
3 7
m 0 0 0 m
m 0 0 0 m
m 0 0 0 m
m 0 0 0 m
ª « « « « ¬
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 301/344
4 8m 0 0 0 m¬
mi Q, 1 d i d 8} P,
B5 = i
i
i 0
a xf
-®¯¦ ai =
1
2
3
m 0 0 0 0
m 0 0 0 0
m 0 0 0 0
ª « « « «
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 302/344
Consider
M1 = ii
i 0a x
f
-®̄¦ ai =
1
2
m
m
0
0
0
0
0
0
ª º« »« »« »« »
« »« »« »« »« »« »« »¬ ¼
with m1, m2 Q}
is a vector subspace of V over Q.
M2 = i
i
i 0
a xf
-®¯¦ ai =
1
2
3
0
0
mm
m
ª º« »« »
« »« »« »« »« »« »
with mi Q; 1 d i d
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 303/344
4
5
6
m
m
m
« »« »« »« »¬ ¼
is a vector subspace of V over Q. Clearly 1MA = M
versa.
Consider
M3 = ii
i 0a x
f
-®̄¦ ai =
1
2
3
0
0
m
m
m
0
0
0
ª º« »« »« »« »
« »« »« »« »« »« »« »¬ ¼
, m1, m2, m3
M3 is also a vector subspace of V over Q and fo
and for every y M3 we see x un y = (0) howeve
M1 + M3 z V however M1 + M2 = V and 1MA =
M1.
Thus we see we can have subspaces in V or
but they need not be the orthogonal complem
over Q.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 304/344
Now we proceed onto define semivector
matrix coefficient polynomials defined over the{0} or Q
+ {0} or R + {0}.
We just describe them in the following.
1dª º
W = i
i
i 0
a xf
-®¯¦ ai = (m1 | m2 | m3 m4) with m j
1 d j d 4} is a super column matrix coefficient
semivector space over the semifield S = Q+ {0{0}).
T =i
ia x-®¯¦ ai =
1 2 3
4 5 6
7 8 9
10 11 12
13 14 15
16 17 18
19 20 21
22 23 24
25 26 27
d d d
d d d
d d d
d d d
d d d
d d d
d d d
d d d
d d d
ª º« »« »« »« »« »« »« »« »« »« »« »« »« »¬ ¼
with d j R +
1 d j d 27} is a super column vector coefficient
semivector space over the semifield S = Z+ {0} (o
or R+ {0}).
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 305/344
or R {0}).
M = i
i
i 0
a xf
-®¯¦ ai =
1 2 3 4 5 6
8 9 10 11 12 13
15 16 17 18 19 20
t t t t t t
t t t t t t
t t t t t t
ª « « « ¬
Q+ {0}, 1 d i d 21} is the super row vector
polynomial semivector space over the semifield S =+
is a super 4 u 8 matrix coefficient polynomial sem
over the semifield Z+ {0}.
C = i
i
i 0
a xf
-®¯¦ ai =
1 2 3 4
7 8 9 10
13 14 15 16
19 20 21 22
25 26 27 28
31 32 33 34
m m m m
m m m mm m m m
m m m m
m m m m
m m m m
ª « « « « « « « « ¬
m j Z+ {0}, 1 d i d 36}
is a super square matrix coefficient polynom
space over the semifield Z+ {0}.
The authors by examples show how subse
direct sum etc looks like.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 306/344
Example 6.55: Let
M = i
i
i 0
a xf
-®¯¦ ai =
1 2 3 4 5 6 7
9 10 11 12 13 14 15
m m m m m m m
m m m m m m m
ª « «
Consider
p(x) =
8 0 1 7 0 3 8 1
0 0 2 0 8 1 0 1
1 5 0 1 0 0 0 1
ª º« »« »« »¬ ¼
+
6 7 0 1 6 0 9 1
0 9 1 2 0 1 6 2
2 0 2 8 1 1 7 3
ª º« »« »« »¬ ¼
x +
1 0 1 8 1 2 0 7
1 2 4 0 3 2 0 0
2 0 3 1 1 0 0 2
ª º« »« »« »¬ ¼
x2
and
q(x) =
0 1 2 1 0 5 7 2
2 0 1 3 7 0 5 1
1 4 8 1 0 0 4 0
ª º« »« »« »¬ ¼
+
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 307/344
2 8 1 3 4 3 3 1
1 0 5 0 1 2 2 2
0 1 4 6 0 0 1 3
ª º« »« »« »¬ ¼
x2 +
0 0 1 3 0 0 6 2ª º« » 3
and
P = i
i
i 0
a xf
-®¯¦ ai =
1 4 5
2 7 8
3 10 11
m 0 0 m m
m 0 0 m m
m 0 0 m m
ª « « « ¬
mi Q+ {0}, 1 d i d 12} M,
T and P are super row vector polynomial coeffic
subspace of M over the semifield Q+ {0};
we see M = T + P with T P =
0 0 0 0
0 0 0 0
0 0 0 0
ª « « « ¬
Further for every x T we have a y P;
with x un y =
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
ª « « « ¬
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 308/344
Example 6.56: Let
f
1 2 3 4
5 6 7 8
9 10 11 12
d d d d
d d d d
d d d d
ª º« »« »« »« »
be a super column vector polynomial coefficientspace (linear algebra) over the semifield S = Z+ {
Now we can define like wise semivector spac
matrices and study those structures.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 309/344
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 310/344
One has to find the uses of these new structurvalue problems, mathematical models, coding theor
element analysis methods.
Further these natural product on matrices worusual product on the real line and the matrix prod
matrices. If the concept of matrices is a an array of n
certainty the natural product seems to be appropriat
course of time researchers will find nice applications
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 311/344
&KDSWHU(LJKW
68**(67('352%/(06
In this chapter we suggest over 100 problems.can be taken up as research problems. These prmakes the reader to understand these new notiothis book.
1. Find some interesting properties enjoyed by
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 312/344
with matrix coefficients.
2. For the row matrix coefficient polynomial s
S[x] = ii
i 0
a x∞
=
-®¯¦ ai = (x1, x2, x3, x4) with x j
(i) Find zero divisors in S[x].(ii) Can S[x] have ideals?
(iii) If the row matrix coefficients are from Z β be in Z × Z × Z × Z?
4. Give some nice properties enjoyed by the sesquare matrix coefficient polynomials.
5. Solve the equation1 11 1
ª º« »¬ ¼
x3 –2 7 81 6 4
ª º« »¬ ¼
= 0
6. Suppose p(x) =
9
2
1
3
7
ª º« »
« »« »« »« »« »¬ ¼
–
8
9
2
3
1
ª º« »
« »« »« »« »« »−¬ ¼
x +
3
7
0
8
1
ª º« »
« »« »« »« »« »¬ ¼
x2. Solve
7. Let p(x) = (1, 2, 3)x3 – (2, 4, 5)x2 + (1, 0, 2)x –Solve for x.Does q(x) = (1, 6, 9)x + (2, 1, 3) divide p(x)?
8. Find the properties enjoyed by the group of squ
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 313/344
coefficient polynomials in the variable x.
9. Let p(x) =2 1 0
1 5 6
§ ·¨ ¸© ¹
+7 8 0
1 1 1
§ ·¨ ¸© ¹
x2 +3
0
§ ¨ ©
Is p(x) solvable as a quadratic equation?
11. Suppose p(x) =9 i
ii 0
a x=¦ where ai ∈ {V3×6 =
matrices with coefficients from Z}, 0 ≤ i ≤cannot be integrated and the resultant coeffbe in V3×6.
12. Prove VR = {(a1, a2, …, a12) | ai ∈ R} has zunder product.
13. Prove V3×3 =1 2 3
4 5 6
7 8 9
a a a
a a a
a a a
-§ ·°¨ ¸®
¨ ¸°¨ ¸© ¹¯
ai ∈ Z, 1 ≤
semigroup under multiplication.(i) Is V3×3 a commutative semigroup?(ii) Find ideals in V3×3.(iii) Can V3×3 have subsemigroups which ar
14. Can VR [x] = ii
i 0
a x∞
=
-®¯¦ ai = (x1, x2, …, x8)
1 ≤ j ≤ 8} a semigroup under product have
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 314/344
1 ≤ j ≤ 8}, a semigroup under product have(i) Find ideals of VR [x]?
(ii) Find subsemigroups which are not ide(iii) Can VR [x] be a S-semigroup?
15. Let V7×7 [x] = ii
i 0
a x∞
=
-®¯¦ ai’s are 7 × 7 matr
17. Distinguish between Q [x] and
VR [x] = ii
i 0
a x∞
=
-®¯¦ ai = (x1, x2, x3, x4); x j ∈ Q;
18. What are the benefits of natural product in mat
19. Prove Vn×n [x] under natural product is a commsemigroup.
20. Prove V3×7 [x] is a commutative semigroup und
product ×n.
21. Prove natural product ×n and the usual product matrices on VR [x] are identical.
22. Show VC [x] under natural product is a semigro
zero divisors.
23. Obtain some nice properties enjoyed by
1a-ª º°« »
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 315/344
VC = 2
m
a
a
°« »°« »
®« »°« »°¬ ¼¯
#
ai ∈ Q; 1 ≤ i ≤ m} under the natu
product, ×n.
24. Show V5×2 = {all 5 × 2 matrices with entri
(v) Can V5×2 have idempotents justify?
25. Show (V3×3, ×n) and (V3×3, ×) are distinct as
(i) Can they be isomorphic?
(ii) Find any other stricking difference betw
26. Can the set of 5 × 5 diagonal matrices withunder the natural product and the usual pro
27. Prove (V2×2, +, ×n) is a commutative ring.
28. Prove (V3×3, +, ×) is a non commutative rin
1 0 0
0 1 0
0 0 1
§ ·¨ ¸¨ ¸¨ ¸© ¹
as unit.
29. Find the differences between a ring of natural product and usual matrix product.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 316/344
30. Prove (V2×7, +, ×n) is a commutative ring w
31. Let S = (V5×2, +, ×n) be a ring.(i) Find subrings of S.
(ii) Is S a Smarandache ring?
(iii) Can S have S-subrings?
33. Distinguish between (VR, +, ×n) and (VR, +, ×).
34. Find the difference between the rings (V3×3,(V3×3, +, ×n).
35. Let M = ( RV+ , +, ×n) be a semiring where
RV+ = {(x1, x2, …, xn) | xi ∈ R+ ∪ {0}, 1 ≤ i ≤ n
(i) Is M a semifield?
(ii) Is M a S-semiring?
(iii) Find subsemiring in M.(iv) Show every subsemiring need not a be S-su
(v) Find zero divisors in M.
(vi) Can M have idempotents?
36. Let P = {VC =
1
2
3
a
a
a
-ª º°« »°« »°« »®
« »°ai ∈ Q+ ∪ {0}; 1 ≤
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 317/344
4
5
a
a
« »°« »
°« »°¬ ¼¯semiring under + and ×n.
(i) Find ideals of P.(ii) Is P a S-semiring?
38. Mention some of the special features semiring of 5 × 8 matrices with + and ×n
from R+ ∪ {0}.
39. Is P =
1
2
3
4
5
x
xx
x
x
-ª º°« »°« »°« »®
« »°« »°« »°¬ ¼¯
xi ∈ R+; 1 ≤ i ≤ 5} ∪
0
00
0
0
- ½ª º° °« »° °« »° °« »® ¾
« »° °« »° °« »° °¬ ¼¯ ¿
under + and ×n?
40. Can M =1 1 1 1 1
a b c d e
a b c d e
-ª º°®« »
¬ ¼°̄a1, b1, c
d, e ∈ Q+} ∪ 0 0 0 0 0
0 0 0 0 0
- ½ª º° °® ¾« »° °¬ ¼¯ ¿
, +, ×n}
a a-ª º
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 318/344
41. Find for S =
1 2
3 4
5 6
7 8
a a
a aa a
a a
-ª º°« »
°« »®« »°« »°¬ ¼¯
ai ∈ Q+ ∪ {0}, 1
the semiring.
42. Let P =
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
a a a aa a a a
a a a a
a a a a
-ª º°« »°« »®« »°« »°¬ ¼¯
ai ∈ Q+ ∪ {0}
1 ≤ i ≤ 16} be a semiring under + and natural p
(i) Is P a S-ring?
(ii) Can P have zero divisors?
(iii) Show P is commutative.
(iv) If ×n replaced by usual matrix product wi
semiring? Justify your claim.
(v) Find S-ideals in P.(vi) Find subsemirings which are not S-subse
43. Let V = {(x1, x2, …, xn) | xi ∈ R, 1 ≤ i ≤ n} F Fi d H (V V)
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 319/344
space over F. Find Hom (V, V).
44. Let P =
1
2
10
x
x
x
-ª º°« »°« »®« »°« »°¬ ¼¯
#xi ∈ Q; 1 ≤ i ≤ 10} be a lin
46. What is the difference between a naturaspecial field and the field?
47. Obtain the special properties enjoyed by Scolumn matrix linear algebra.
48. Obtain the special and distinct features of 3 × 3 matrix linear algebra.
49. Find differences between Smarandache veSmarandache special strong vector spaces.
50. Let P = 1 2
3 4
a aa a
-ª º°®« »
¬ ¼°̄ai ∈ R, 1 ≤ i ≤ 4} be t
matrix of natural special Smarndache field.
(i) What are the special properties enjoyed
(ii) Can P have zero divisors?
51. Obtain some interesting properties about column matrix vector spaces constructed ov
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 320/344
52. Find some applications of S-special strong
linear algebras constructed over Q.
53. Define some nice types of inner products ousing the natural product ×n.
54 Can linear functionals be defined on S sp
F3×12 =1 2 12
13 14 24
25 26 36
x x ... xx x ... x
x x ... x
-ª º°« »®« »°« »¬ ¼¯
xi ∈ Q, 1 ≤ i ≤
(i) Find a basis for V.
(ii) What is the dimension of V over F3×12?(iii) Write V as a direct sum of subspaces.(iv) Write V as a pseudo direct sum of subspa
56. Let V be a S-special strong vector space of n ×
over the S-field FC
of n × n matrices with elethe field Q.(i) Find a basis for V.(ii) Write V as a direct sum of subspaces.(iii) Write V as a pseudo direct sum of subspa(iv) Find a linear operator on V.
(v) Does every subspace W of V have W
⊥
?(vi) Write V as W + W⊥;
57. Obtain some interesting properties about subspaces.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 321/344
58. Find some interesting properties related witrow matrix linear algebras.
59. Study the special properties enjoyed by S-spm × n matrix linear algebras (m ≠ n).
(i) Find basis for S.(ii) Write S as a direct sum of subsemivec(iii) Write S as a pseudo direct sum of subs
spaces.(iv) Can S be a semilinear algebra?
61. Obtain some interesting properties enjoyedspace of column matrices V over the semi{0}.
62. Enumerate the special properties en
semivector space of m × n matrices (msemifield F = Q+ ∪ {0}.
63. Bring out the differences between the semcolumn matrices over Q+ ∪ {0} and vcolumn matrices over Q.
64. Find some special properties enjoyed by sof m × n matrices over the semifield Z+ ∪ {
1 2 3 4 5a a a a a-ª º°
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 322/344
65. Let V =
1 2 3 4 5
6 7 10
11 12 15
16 17 20
a a a a a
a a ... ... aa a ... ... a
a a ... ... a
-ª º°« »°« »®« »°« »°¬ ¼¯
ai ∈ Z
20} be a semivector space over the semi{0}.
(v) Write V = W ⊕ W⊥
, W⊥
the orthogonal comof W.
66. Let SCF =
1
2
3
4
5
6
a
a
a
a
a
a
-ª º°« »°« »°« »°
« »®« »°« »°« »°« »°¬ ¼¯
ai ∈ Q, 1 ≤ i ≤ 6, ×n} be a sem
(i) Find ideals in SCF .
(ii) Can SCF have subsemigroups which are not
(iii) Prove SCF has zero divisors.
(iv) Find units in SCF .
(v) Is SCF a S-semigroup?
67. Let SRF = {(a1 a2 | a3 a4 | a5) | ai ∈ Z, 1 ≤ i ≤
semigroup.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 323/344
(i) Find subsemigroups which are not ideals (ii) Find zero divisors in S
RF .
(iii) Can SRF have units?
(iv) Is x = (1, –1 | 1 –1 | –1) a unit in SRF .
(i) Find units in S
5 3F × .
(ii) Is S5 3F × a S-semigroup?
(iii) Can S5 3F × have S-subsemigroups?
(iv) Can S5 3F × have S-ideals?
(v) Does S5 3F × have S-zero divisors/
(vi) Can S5 3F × have S-idempotents?
(vii) Show S5 3F × have only finite number of
69. Let S4 4F × =
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
a a a aa a a a
a a a a
a a a a
-ª º°« »
°« »®« »°« »°¬ ¼¯
ai ∈ Q,
semigroup of super square matrices.
(i) Find zero divisors in S4 4F × .
(ii) Can S4 4F × have S-zero divisors?
(iii) Can SF have S-idempotents?
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 324/344
(iii) Can 4 4F × have S idempotents?
(iv) Find the main complement of
3 01 0
0 3 1
0 1
ª « « « « ¬
( ) I SF S i ?
(i) Find ideal of S3 7F × .(ii) Is S
3 7F × a S-semigroup?
(iii) Can S3 7F × have S-ideals?
(iv) Show S3 7F × can have only finite number o
idempotents.
(v) Show S3 7F × has no units.
71. Obtain some interesting properties about ( SCF , ×
72. Find some applications of the semigroup ( SmF
×n).
73. Find the difference between ( Sn nF × , ×) and ( S
n nF ×
74. Find some special and distinct features enjoyed
75. Prove { Sn mF × , n ≠ m, +, ×n} is a commutat
infinite order.
6 S 1 2 3 4 5a a a a a-ª º°® Q 1
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 325/344
76. Let S2 5F × = 1 2 3 4 5
6 7 8 9 10a a a a a
ª º°®
« »¬ ¼°̄ai ∈ Q, 1
×n} be a ring of super row vectors.
(i) Find ideals in S2 5F × .
(ii) Is S2 5F a S-ring?
77. Let S8 3F × =
1 2 3
4 5 6
7 8 9
10 11 12
13 14 15
16 17 18
19 20 21
22 23 24
a a aa a a
a a a
a a a
a a a
a a a
a a a
a a a
-ª º°« »°« »°« »°« »°« »®« »
°« »°« »°« »°« »°« »¬ ¼¯
ai ∈ Q, 1 ≤ i
a super column vector ring.
(i) Prove S8 3F × has zero divisors.(ii) Prove S
8 3F × has units.
(iii) Can S8 3F × have S-units?
(iv) Is S8 3F × a S-ring?
(v) Prove S
8 3F
×
has idempotents?
1
2
3
a
a
a
-ª º°« »°« »°« »°« »
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 326/344
78. Let SCF =
4
5
6
7
8
aa
a
a
a
°« »°« »°« »°« »®
« »°« »°« »°« »°
ai ∈ R, 1 ≤ i ≤ 10, +,
(iii) Can SCF have S-idempotents?(iv) Can S
CF have S-units?
79. Let S3 3F × =
1 2 3
4 5 6
7 8 9
a a a
a a a
a a a
-ª º°« »®« »°« »¬ ¼¯
ai ∈ Q, 1 ≤ i ≤ 8,
square super matrix ring.(i) Prove S
3 3F × is a commutative ring.
(ii) Find ideals in S3 3F × .
(iii) IsS
3 3F × a S-ring?(iv) Show S
3 3F × has only finite number of idemp
(v) Can S3 3F × have S-ideals?
80. Enumerate some special features enjoyed by s
ringsS
CF (orS
RF orS
n nF × orS
n mF × ; m ≠ n).
81. Find some applications of the rings mentioned(80).
82 Prove R = {(a1 | a2 | | an) | ai ∈ Q+ ∪ {0} 1
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 327/344
82. Prove R = {(a1 | a2 | … | an) | ai ∈ Q ∪ {0}, 1
semigroup under +.
(i) Can R have ideals?
(ii) Can R have S-zero divisors?
84. Let T =
1
2
3
4
5
aa
a
a
a
-ª º°« »°« »°« »®
« »°« »°« »°¬ ¼¯
ai ∈ Z+ ∪ {0}, 1 ≤ i
semigroup under ×n.(i) Prove T is a commutative semigroup
(ii) Can T have S-zero divisors?
(iii) Show T can have only finite number o
idempotents.
(iv) Show T can have no units.
(v) Can T have S-ideals?
85. Let W =1 2 3 4 5
6 7 8 9 10
11 12 13 14 15
a a a a a
a a a a a
a a a a a
-ª º°« »®« »°« »¬ ¼¯
ai ∈
≤ 15, ×n} be a semigroup.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 328/344
≤ 15, ×n} be a semigroup.
(i) Find the number of idempotents in W.(ii) Is W a S-semigroup?(iii) Find units in W.(iv) Show all elements are not units in W.
a a a a a a-ª º
(ii) Does M contain S-zero divisors?(iii) Prove M has only finite number of idempot
(iv) Can M have S-idempotents?
(v) Can M have S-units?
87. Let M = S2 3F × = 1 2 3
4 5 6
a a a
a a a
-ª º°®« »
¬ ¼°̄ai ∈ Z+ ∪ {0
+, ×n} be a semiring.
(i) Prove M is not a semifield.(ii) Find subsemirings in M.
88. Obtain some interesting properties enjoyed super matrix semirings with entries from Q+ ∪
89. Distinguish between a super square matrix super square matrix semiring.
1 6a a-ª º°« »
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 329/344
90. Let P =2 7
3 8
4 9
5 10
a aa a
a a
a a
°« »
°« »°« »®« »°« »°« »°¬ ¼¯
ai ∈ R+ ∪ {0}, 1 ≤ i ≤ 10,
semiring.
91. Let T =
1 2 3 4 5 6
8 9
15 16
22 23
29 30
36 37
43 44
a a a a a a aa a ... ... ... ... a
a a ... ... ... ... a
a a ... ... ... ... a
a a ... ... ... ... a
a a ... ... ... ... a
a a ... ... ... ... a
-ª °« °« °« °« °®« °«
°« °« °«
¬ °̄
{0}, 1 ≤ i ≤ 49, +, ×n} be a semiring.
(i) Show T has only finite number of idem(ii) Find zero divisors of T.
(iii) Find idempotents of T.
(iv) Can T have S-zero divisors?
(v) Is T a S-semiring?
92. Find some interesting properties of semirings.
a a a aª º 0 0ª
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 330/344
93. Let M = {
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
a a a a
a a a a
a a a a
a a a a
ª º« »« »« »« »¬ ¼
∪
0 0
0 0
0 0
0 0
ª « « « « ¬
94. Can P =
1
2
3
4
5
6
aa
a
a
a
a
-ª º°« »°« »°« »°
« »®« »°« »°
« »°« »°¬ ¼¯
∪
00
0
0
0
0
ª º« »« »« »« »« »« »
« »« »¬ ¼
where ai ∈ Q+, 1 ≤ i ≤ 6
a semifield?
95. Is T = {(a1 | a2 a3 | a4 a5 a6) ∪ (0 | 0 0 | 0 0 0) | i ≤ 6, +, ×n} a semifield?
Can T have subsemifields?
96. Let W =
1 2 3 4
5 6 7 8
9 12
13 16
17 20
21 24
25 28
a a a a
a a a a
a . . aa . . a
a . . a
a . . a
a a
-ª º°« »°« »
°« »°« »°« »°« »®
« »°« »°« »°
∪
0 0 0
0 0 0
0 0 00 0 0
0 0 0
0 0 0
0 0 0
ª « «
« « « « « « «
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 331/344
25 28
29 32
33 36
a . . a
a . . a
a . . a
°« »°« »°« »°¬ ¼¯
0 0 0
0 0 0
0 0 0
« « « ¬
ai ∈ R+, 1 ≤ i ≤ 36, +, ×n} be a semifield of sut Fi d b ifi ld f W C
99. Let X =
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
a a a aa a a a
a a a a
a a a a
-ª º°« »°« »®« »°« »°¬ ¼¯
where ai ∈
≤ 16} be a super square matrix linear asemifield S = R+ ∪ {0}.
(i) Find a basis of X over S.
(ii) Is X finite dimensional?
(iii) Write X as a direct sum of semivector
(iv) Write X as a pseudo direct sum
subspaces.
(v) Let V ⊆ X be a subspace find V⊥ so t
a complement?
100. Let V =1 2 3
4 5 6
7 8 9
a a a
a a a
a a a
-ª º°« »®« »°« »¬ ¼¯
ai ∈ Q+ ∪ {0},
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 332/344
¯super matrix semivector space over the se∪ {0}.
(i) Can W =1 2 3a a a
a a a
-ª º°« »®« » ai ∈ Z
101. Let V =
1 2 3 4 5 6
7 8 9 10 11 12
13 14 15 16 17 18
19 20 21 22 23 24
a a a a a aa a a a a a
a a a a a a
a a a a a a
-ª º°« »°« »®« »°« »°« »¬ ¼¯
ai ∈
1 ≤ i ≤ 24} be a super matrix semilinear algebZ+ ∪ {0}.
(i) Is V finite dimensional?
(ii) Find subspaces of V so that V can be w
direct sum of super matrix semivector subs(iii) Write V as a pseudo direct sum of su
semilinear algebra.
(iv) Let M =
1 6 7 8
2 9 10 11
3 12 13 14
4 5
a 0 0 a a a
a 0 0 a a aa 0 0 a a a
0 a a 0 0 0
-ª º°« »°« »®« »°« »°« »¬ ¼¯
{0}, 1 ≤ i ≤ 14} ⊆ V be a super matrixsubalgebra of V.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 333/344
g
a) How many complements exists for M?b) Write down the main complement of M.
(iii) Can V have linearly independent enumber (cardinality) is greater than thof the basis of V over S?
103. Let M =
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
17 18 19 20
21 22 23 24
25 26 27 28
29 30 31 32
33 34 35 36
37 38 39 40
a a a a
a a a a
a a a a
a a a a
a a a a
a a a a
a a a aa a a a
a a a a
a a a a
-ª º°« »
°« »°« »°« »°« »°« »°
« »®« »°« »
°« »°« »°« »°« »°« »°¬ ¼¯
where ai ∈
i ≤ 40} be a super column vector semiline
the semifield S = Z+ ∪ {0}.
(i) Find a basis for M over S.
(ii) Write M as a pseudo direct sum of
vector semilinear subalgebra over S.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 334/344
g
(iii) Write M as a direct sum of super
semilinear subalgebras over S.
(iv) Write M = W + W⊥ where W⊥ is
l t f W
(ii) Let M = ii
i 0
a x∞
=
-®¯¦ ai = (0 0 | m1 m2 m3 |
m2, m3 ∈ Q+ ∪ {0}} ⊆ P be a super row linear subalgebra over S. Find M⊥ so thM⊥.
(iii) Let T = ii
i 0
a x∞
=
-®¯¦ ai = (d1 d2 | 0 0 0 | d3
Z+ ∪ {0}, 1 ≤ j ≤ 4} ⊆ P be a semi sublinof P over S. Can we find a orthogonal comT of P over S so that T + T⊥ = P?
105. Let S = ii
i 0
a x∞
=
-®¯¦ ai =
1
2
3
4
5
6
7
8
9
d
d
d
d
dd
d
d
d
ª º« »« »« »« »« »
« »« »« »« »« »« »« »
d j ∈ Z+ ∪ {0}, 1
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 335/344
9
10
11
d
d
« »« »« »« »¬ ¼
a super column matrix semilinear algebrasemifield F = Z+ ∪ {0}.
106. Let T = ii
i 0
a x∞
=
-®¯¦ ai =
1 2 3
4 5 6
7 8 9
10 11 12
13 14 15
16 17 18
19 20 21
22 23 24
25 26 27
28 29 30
31 32 33
m m mm m m
m m m
m m m
m m m
m m mm m m
m m m
m m m
m m mm m m
ª º« »« »« »« »« »« »
« »« »« »« »« »« »« »« »« »« »¬ ¼
1 ≤ i ≤ 33} be a super column vector semover S = Z+ ∪ {0}.(i) Find a basis for T over S.
(ii) Can T have more than one basis?(iii) Write T as direct sum.
(iv) Write T as pseudo direct sum.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 336/344
107. Let M = ii
i 0
d x∞
=
-®¯¦ di =
1 2 3
5 6 7
9 10 11
13 14 15
17 18 19
a a aa a a
a a a
a a a
a a a
ª « « « « « « «
108. Obtain some unique properties enjoyed by sucoefficient polynomial rings.
109. Find some special features of super matrixpolynomial semivector spaces over the semifiel
{0}.
110. Describe any special feature enjoyed by sucoefficient polynomial semivector space semifield S = Z+ ∪ {0}.
111. Give some applications of super matrix polynomial semilinear algebras defined over a s
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 337/344
)857+(55 ($',1*
1. Abraham, R., Linear and Multilinear A
Benjamin Inc., 1966.
2. Albert, A., Structure of Algebras, Colloq. P
Math. Soc., 1939.3. Gel'fand, I.M., Lectures on linear algeb
New York, 1961.
4. Greub, W.H., Linear Algebra, Fourth EdVerlag, 1974.
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 338/344
5. Halmos, P.R., Finite dimensional vector Nostrand Co, Princeton, 1958.
6. Harvey E. Rose, Linear Algebra, Bir K2002.
7. Herstein I.N., Abstract Algebra, John Wiley,
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 339/344
,1'(;
D
Derivatives of matrix coefficient polynomials, 21-9
I
Integral of matrix coefficient polynomials, 26-33
L
Linear algebra of super column matrices, 225-9
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 340/344
Linear algebra of super row matrices, 225-9Linear algebra under natural product of matrices, 91-
M
Matrix coefficient polynomials, 8
P
Polynomial with matrix coefficients, 8Polynomials with column matrix coefficients, 11-35Polynomials with row matrix coefficients, 11-35
R
Ring of column matrices under natural product, 63-8Ring of row matrix coefficient polynomials, 16-9
S
Semifield of matrices under natural product, 76-9Semifields, 7Semigroup of row matrix coefficient polynomials, 16-8Semigroup of super column vector, 198-204Semigroup of super row vectors, 198-200Semiring of matrices under natural product, 75-80Semirings, 7
Semivector spaces, 7S-ideal, 7Smarandache linear algebra, 10Smarandache semigroup under natural product, 42-9Smarandache semigroup, 7Smarandache semiring, 77-83Smarandache subsemigroup, 46-53
S d h b i i 77 83
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 341/344
Smarandache subsemiring, 77-83Smarandache vector spaces, 10Special column matrix S-field, 100-4S-ring of matrices under natural product, 70-5S-rings, 7S-Special strong column matrix vector space, 103-7
Super matrix semigroup under natural product, 182-Super row matrix coefficient polynomials, 252-5Super row matrix, 8-9, 163-6Super row vector semiring, 212-6Super row vector, 9, 163-8Super square matrix semiring, 212-7Super square matrix, 9, 163-9
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 342/344
$%2877+($87+256
'U:%9DVDQWKD.DQGDVDP\ LV DQ $VVRFLDWH 3URI
'HSDUWPHQW RI 0DWKHPDWLFV ,QGLDQ ,QVWLWXWH RI0DGUDV &KHQQDL ,Q WKH SDVW GHFDGH VKH KDV JXLGH
VFKRODUV LQ WKH GLIIHUHQW ILHOGV RI QRQDVVRFLDWLY
DOJHEUDLF FRGLQJ WKHRU\ WUDQVSRUWDWLRQ WKHRU\ IX]]\DSSOLFDWLRQV RI IX]]\ WKHRU\ RI WKH SUREOHPV IDFHGLQGXVWULHV DQG FHPHQW LQGXVWULHV 6KH KDV WR KHU
UHVHDUFK SDSHUV 6KH KDV JXLGHG RYHU 06F D
SURZHFWV6KHKDVN RU5HGLQFROODERUDWLRQSURZHFWVN LW
6SDFH2HVHDUFK1UJDQL]DWLRQDQGN LWKWKH7DPLO%DGX
&RQWURO 6RFLHW\ 6KH LV SUHVHQWO\ N RU5LQJ RQ D UHVH
IXQGHG E\ WKH *RDUG RI 2HVHDUFK LQ %XFOHD RYHUQPHQWRI,QGLD7KLVLVKHU
UGERR5
1Q ,QGLDV 9WK ,QGHSHQGHQFH 'D\ 'U.DV
FRQIHUUHG WKH (DOSDQD &KDN OD $N DUG IRU &RXUDJH QWHUSULVH E\WKH 6WDWH RYHUQPHQW RI 7DPLO %DGXLQ
RI KHU VXVWDLQHG ILJKW IRU VRFLDO ZXVWLFH LQ WKH ,QGLDQ
7HFKQRORJ\,,70DGUDVDQGIRUKHUFRQWULEXWLRQWRP
7KH DN DUG LQVWLWXWHG LQ WKH PHPRU\ RI ,QGLDDVWURQDXW (DOSDQD &KDN OD N KR GLHG DERDUG 6S
&ROXPELD FDUULHG D FDVK SUL]H RI ILYH OD5K UXSHHV
SUL]HPRQH\IRUDQ\,QGLDQDN DUGDQGDJROGPHGDO
6KHFDQEHFRQWDFWHGDW YDVDQWKD5DQGDVDP\JPDLOFR8HE6LWHKWWS::PDWLLWPDFLQ:KRPH:N EY:SXEOLF;KWPO
RUKWWS::N N N YDVDQWKDLQ
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 343/344
'U )ORUHQWLQ 6PDUDQGDFKH LV D 3URIHVVRU RI 0DW
WKH <QLYHUVLW\ RI%HN 0H=LFRLQ<6$ >HSXEOLVKHG RY
DQG A99 DUWLFOHV DQG QRWHV LQ PDWKHPDWLFV SK\VLFV
SV\FKRORJ\ UHEXV OLWHUDWXUH ,Q PDWKHPDWLFV KLV UHQXPEHU WKHRU\ QRQ XFOLGHDQ JHRPHWU\ V\QWKHWLF
DOJHEUDLF VWUXFWXUHV VWDWLVWLFV QHXWURVRSKLF ORJL
8/3/2019 Natural Product Xn on Matrices, by W. B. Vasantha Kandasamy, Florentin Smarandache
http://slidepdf.com/reader/full/natural-product-xn-on-matrices-by-w-b-vasantha-kandasamy-florentin-smarandache 344/344