Linear Algebra 022315

7/24/2019 Linear Algebra 022315

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03/04/08 Revised by D.H. Chen 2


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irr!r "#age $.r.%. &rin'i(al Diag!nal


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03/04/08 Revised by D.H. Chen )

"den%i%y a%ri*


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03/0)/12 Revised by D.H. Chen 6

+l%i(li'a%i!n


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2 - 3 2 - 33 - 3

3 - 3 A 2- 3

3 - 3 3 - 1 3 - 1

i e Re +ire#en%2nd inde* ! %he 1s% #a%ri* #+s% be %hesa#e as %he 1s% inde* !

%he 2nd

#a%ri*


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02/23/1) Revised by D.H. Chen 8

C! a'%!r

in!r 5 is %he De%er#inan% ! a +b a%ri* by dele%ing %hei %h r!$ and j %h'!l+#n r!# %he +ll #a%ri*.


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De%er#inan%A 'hara'%eris%i' s'alar val+e9 ! a #a%ri*

Cr!ss &r!d+'%


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03/0)/12 Revised by D.H. Chen 12

AA1;"


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de% A 0de% A ; 0 de% A ; 0


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" A is n!n sing+lar=

i.e.= de% A 0 b 0 n!n h!#!gene!+s9? %hen %here is a

+ni +e se% ! n!n%rivial s!l+%i!n= i.e.= * 0as sh!$n in 'ase b9

b ; 0 h!#!gene!+s9 ? %hen %here is a%rivial s!l+%i!n= i.e.= * ; 0

03/17/12 Revised by D.H. Chen 1)


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Revised by D.H. Chen 16

a%ri* A is sing+lar i de% A ; 0

A n!n%rivial s!l+%i!n se% in $hi'h%here e*is%s a% leas% !ne @ree ariable

H!#!gene!+s Linear ys%e#

l


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03/04/08 Revised by D.H. Chen 1,

Ba+ssian li#ina%i!n


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a'5 +bs%i%+%i!n


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"nverse A1

riginala%ri* A

De%er#inan%|A |

de%A;4E3E .)9E 1.83339


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3 :y(es C!n%.9 #;n Algebrai' ns s!lved $i%h Fe$%!nGs

#e%h!d?i*ed Di eren%ial/Algebrai' ns

s!lved byBearGs #e%h!d #>n Regressi!n Analysis s!lved $i%har +ard%Gs #e%h!d

# < n (%i#i a%i!n s!lved $i%hLinear&r!gra##ing L&9 !r +''essive K+adra%i'&r!gra##ing K&9


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General Multi-Linear Model Y = a 1V1+ a 2V2 +a 3V3 + ....a nVn + C

where a i = weighting factor

Vi = predictor variable

C = constant


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Linear Regression (3)

:a5e %he e*a#(le ! %he d#is%er va(!r (ress+re e +a%i!nLn & ; a b/ : ; a E1 bE 1/ :

as an e*a#(le Le%Gs say $e have 100 da%a (!in%s= %hen100= n ;2=x ; a= b9T= A is 100*29= AT is

2*1009= A:

EAE is 2*29= b is 100*19= AT* b is 2*19= s! $e s!lve !r 2 e ns and2 +n5n!$ns in %he n!r#al e +a%i!n.


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03/04/08 Revised by D.H. Chen 3,

Mha% are A=x=b= A: N


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Linear Regression (4)

'odel uation in 4'5 is a similaro"er-determined linear system (mxn!m n)

A* '6 = where = 78-86 open (1) ow we ha"e an exa t (nxn) system for

regression to sol"e for '6 (future

'6 mo"es) AT*A* '6 = A T*


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03/17/12Revised by D.H. Chen 37

; ln a E 1 b E :


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R- squared O Pi Q i92

R 2 ; 1 Q SSSSSO Pi QT92

Mhere Pi Q T9 ; devia%i!n ! %hei%h !bserva%i!n r!# %he !verall #ean

P i Q i9 ; di eren'e be%$een %he (redi'%edand %he a'%+al da%a !r %hei%h!bserva%i!n.


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9-s uared for inear 'odel

The Goodness of %it or the xplained6ariation #

Also /nown as 5oeffi ient of 4etermination 9;< = 77 due to regression Total 77

orre ted for the mean = (>i? > );< (>i > );i > );< = (>i? > );< @ (>i >i?);i = data > = mean >i? = predi tion


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Regressi!n !dels 29 P ; a b E - P ; a b E - ' E -U2

"s %his a linear #!delN Pes. :he e*(ressi!n is linear in (ara#e%ers a= b= '

L!g & ; A /: "s %his a linear #!delN Pes. :he e*(ressi!n is linear in

(ara#e%ers A L!g & and 1/: $ill be vie$ed as 5n!$n da%a

L!g & ; A / : C9 "s %his a linear #!delN F!. :he e*(ressi!n is n!n linear

in (ara#e%ers A= = and C.


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Regressi!n !dels 39

r A9 ; 5C / 1 C A 9

"s %his a linear #!delN F!. :his is n!nlinear !r%he (ara#e%ers 5= == and.


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03/04/08 Revised by D H Chen 4

Regressi!n *a#(le

Typical Tent y =

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Linear Algebra 022315