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In this paper we will present a new method to calculate of nxn order determinants. This method is based on Dodgson – Chio’s condensation method, but the priority of this method compared with Dodgson – Chio’s and minors method as well is that those methods decreases the order of determinants for one, and this new method automatically affects in reducing the order of determinants in 2nd order
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Read! in the name of thy GOD Who created (every thing),
Created man from a clot of blood (in the womb),
Read! and thy GOD is most Generous,
Who taught (the men) by the pen,
Taught man what he knew not.
© Copyrighted International Conference On Applied Analysis and Algebra, Yildiz Technocal University, 20 - 24 june 2012, Istanbul - TURKEY
Armend SalihuUniversity of Prishtina – Faculty of Electrical and Computer Engineering [email protected]+377 45 562 221
New method to calculate determinants of nxn matrix, by reducing determinants to 2nd order
Armend Salihu
Department of Telecommunication, Faculty of Electrical and Computer Engineering,
University of Prishtina, Prishtine, KosovoEmail Address: [email protected]
Abstract
In this paper we will present a new method to calculate of nxn order determinants. This method is based on Dodgson – Chio’s condensation method, but the priority of this method compared with Dodgson – Chio’s and minors method as well is that those methods decreases the order of determinants for one, and this new method automatically affects in reducing the order of determinants in 2nd order.
Keywords: New method to calculate determinants of nxn matrix.
© Copyrighted International Conference On Applied Analysis and Algebra, Yildiz Technocal University, 20 - 24 june 2012, Istanbul - TURKEY
Armend SalihuUniversity of Prishtina – Faculty of Electrical and Computer Engineering [email protected]+377 45 562 221
1. Introduction
Let A be a nxn matrix:
DEFINITION 1: A determinant of order n, or size nxn (see: [5], [6], [7], [8]), is the sum:
ranging over the symmetric permutation group Sn, where:
© Copyrighted International Conference On Applied Analysis and Algebra, Yildiz Technocal University, 20 - 24 june 2012, Istanbul - TURKEY
Armend SalihuUniversity of Prishtina – Faculty of Electrical and Computer Engineering [email protected]+377 45 562 221
1.2. Chio’s Condensation method
Chio’s Condensation is a method for evaluating an nxn determinant in terms of (n-1)x(n-1) determinants (see: [1], [2]):
© Copyrighted International Conference On Applied Analysis and Algebra, Yildiz Technocal University, 20 - 24 june 2012, Istanbul - TURKEY
Armend SalihuUniversity of Prishtina – Faculty of Electrical and Computer Engineering [email protected]+377 45 562 221
1.2. Dodgson Condensation method
The Dodgson’s condensation method is a method, which determinants of the order nxn expansion in determinant of the (n −1)x(n −1) order, than (n − 2)x(n − 2) order and so one (see: [3]).
© Copyrighted International Conference On Applied Analysis and Algebra, Yildiz Technocal University, 20 - 24 june 2012, Istanbul - TURKEY
Armend SalihuUniversity of Prishtina – Faculty of Electrical and Computer Engineering [email protected]+377 45 562 221
1.2. Dodgson Condensation method
Dodgson Condensation Method algorithm can be described in the following 4 steps (see: [10]):
1. Let A be the given nxn matrix. Arrange A so that no zeros occur in its interior. An explicit definition of interior would be all ai,j with i, j ≠ 1, n. We can do this using any operation that could normally perform without changing the value of determinant, such as adding a multiple of one row to another.
2. Create an (n-1)x(n-1) matrix B, consisting of the determinants of every 2x2 sub matrix of A, we write
3. Using this (n-1)x(n-1) matrix, perform step 2 to obtain an (n-2)x(n-2) matrix C. Divide each term in C by the corresponding term in the interior of A.
4. Let A = B, and B = C. Repeat step 3 as necessary until the 1x1 matrix is found; its only entry is the determinant.
© Copyrighted International Conference On Applied Analysis and Algebra, Yildiz Technocal University, 20 - 24 june 2012, Istanbul - TURKEY
Armend SalihuUniversity of Prishtina – Faculty of Electrical and Computer Engineering [email protected]+377 45 562 221
2. New method
This method is based on Dodgson and Chio's method, but the deference between them is that this new method is resolved by calculating 4 unique determinants of (n – 1)x(n – 1) order (which can be derived from determinants of nxn order, if we remove first row and first column or first row and last column or last row and first column or last row and last column, elements that belongs to only one of unique determinants we should call them unique elements), and one determinant of (n – 2)x(n – 2) order which is formed from nxn order determinant with elements a ij with i, j ≠ 1, n, on condition that the determinant of (n – 2)x(n – 2) ≠ 0.
© Copyrighted International Conference On Applied Analysis and Algebra, Yildiz Technocal University, 20 - 24 june 2012, Istanbul - TURKEY
Armend SalihuUniversity of Prishtina – Faculty of Electrical and Computer Engineering [email protected]+377 45 562 221
2. New method
Theorem 1: Every determinant of nxn (n>2) order can be reduced into 2x2 order determinant, by calculating 4 determinants of (n-1)x(n-1) order, and one determinant of (n-2)x(n-2) order, on condition that (n-2)x(n-2) order determinants to be different from zero.
Ongoing is presented a scheme of calculating the determinants of nxn order according to this formula:
The |B| is (n-2)x(n-2) order determinant which is the interior determinant of determinant |A| while |C|, |D|, |E| and |F| are unique determinant of (n-1)x(n-1) order, which can be formed from nxn order determinants.
© Copyrighted International Conference On Applied Analysis and Algebra, Yildiz Technocal University, 20 - 24 june 2012, Istanbul - TURKEY
Armend SalihuUniversity of Prishtina – Faculty of Electrical and Computer Engineering [email protected]+377 45 562 221
2. New method (cont.)
Proof: Lets be n=4, and we will prove that the same result we can achieve when we calculate this determinant according to the above scheme:
© Copyrighted International Conference On Applied Analysis and Algebra, Yildiz Technocal University, 20 - 24 june 2012, Istanbul - TURKEY
Armend SalihuUniversity of Prishtina – Faculty of Electrical and Computer Engineering [email protected]+377 45 562 221
2. New method (cont.)
© Copyrighted International Conference On Applied Analysis and Algebra, Yildiz Technocal University, 20 - 24 june 2012, Istanbul - TURKEY
Armend SalihuUniversity of Prishtina – Faculty of Electrical and Computer Engineering [email protected]+377 45 562 221
2. New method (cont.)
© Copyrighted International Conference On Applied Analysis and Algebra, Yildiz Technocal University, 20 - 24 june 2012, Istanbul - TURKEY
Armend SalihuUniversity of Prishtina – Faculty of Electrical and Computer Engineering [email protected]+377 45 562 221
2. New method (cont.)
Based on this we can outcome to the result: all combinations from which
does not contain one of combinations from |B| determinant, and does not contain one of unique elements, as a result of crossed multiplication, they should be eliminated between each other, while other combinations which contain one of combinations from |B| determinant, extract as common elements and after divided by determinant |B| we get the result of the given determinant.
© Copyrighted International Conference On Applied Analysis and Algebra, Yildiz Technocal University, 20 - 24 june 2012, Istanbul - TURKEY
Armend SalihuUniversity of Prishtina – Faculty of Electrical and Computer Engineering [email protected]+377 45 562 221
2. New method (cont.)
Example: let be the 5 order determinant:
The same result we can achieve even by calculating this determinant in other methods.
© Copyrighted International Conference On Applied Analysis and Algebra, Yildiz Technocal University, 20 - 24 june 2012, Istanbul - TURKEY
Armend SalihuUniversity of Prishtina – Faculty of Electrical and Computer Engineering [email protected]+377 45 562 221
References
[1] F. Chi´o, M´emoire sur les fonctions connues sous le nom de r´esultantes ou de d´eterminants, Turin: E. Pons, 1853.
[2] H. Eves, Chio’s Expansion, §3.6 in Elementary Matrix Theory, New York: Dover, (1996), 129–136.
[3] C. L. Dodgson, Condensation of Determinants, Being a New and Brief Method for Computing their Arithmetic Values, Proc. Roy. Soc. Ser. A, 15(1866), 150–155.
[4] H. Eves, An Introduction to the History of Mathematics, pages 405 and 493, Saunders College Publishing, 1990.
[5] E. Hamiti, Matematika 1, Universiteti i Prishtines: Fakulteti Elektroteknik, Prishtine, (2000), 163–164.
[6] S. Barnard and J. M. Child, Higher Algebra, London Macmillan LTD New York, ST Martin’s Press (1959), 131.
© Copyrighted International Conference On Applied Analysis and Algebra, Yildiz Technocal University, 20 - 24 june 2012, Istanbul - TURKEY
Armend SalihuUniversity of Prishtina – Faculty of Electrical and Computer Engineering [email protected]+377 45 562 221
[7] R. F. Scott, The theory of determinants and their applications, Ithaca, New York: Cornell University Library, Cambridge: University Press, (1904), 3–5.
[8] W. L. Ferrar, Algebra, A Text-Book of Determinants, Matrices, and Algebraic Forms, Second edition, Fellow and tutor of Hertford College Oxford, (1957), 7.
[9] Q. Gjonbalaj, A.Salihu, Computing the determinants by reducing the order by four, Applied Mathematics E-Notes, 10(2010), 151 – 158
[10] http://en.wikipedia.org/wiki/Dodgson_condensation
[11] D. Hajrizaj, New method to compute determinant of a 3x3 matrix, International Journal of Algebra, Vol. 3, 2009, no. 5, 211 – 219
[12] H. Teimoori, E. Sarijloo, A. Amiri and B. Bayat, A New Parallel Algorithm For Evaluating The Determinant of A Matrix of Order n, Islamic Azad University of Zanja, Iran University of Science and Technology, Institute for Advanced studies in Basic Science. Zanja Iran
References
© Copyrighted International Conference On Applied Analysis and Algebra, Yildiz Technocal University, 20 - 24 june 2012, Istanbul - TURKEY
Armend SalihuUniversity of Prishtina – Faculty of Electrical and Computer Engineering [email protected]+377 45 562 221
Someon will ask …
… was Armend boring you?
Ask him any question, I like to challenge
him. L.M.