GENERALIZED MATRIX MULTIPLICATION AND ITS SOME APPLICATION. 1. Introduction

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1 FACTA UNIVERSITATIS (NIŠ) Ser Math Inform Vol, No 5 (7), GENERALIZED MATRIX MULTIPLICATION AND ITS SOME APPLICATION Osman Keçilioğlu and Halit Gündoğan Abstract In this paper, a generalized matrix multiplication is defined in R m,n R n,p by using any scalar product in R n, where R m,n denotes set of matrices of m rows and n columns With this multiplication it has been shown that R n,n is an algebra with unit By considering this new multiplication we define eigenvalues and eigenvectors of square n n matrix A A special case is considered and generalized diagonalization is also introduced Keywords: Generalized matrix multiplication; inner product; eigenvector; eigenvalue Introduction The sign matrix is required for the definition of special matrices when the Euclid inner product is used for the matrix multiplication in Lorentz space [5 Ergin [ obtained the rotations in Lorentz plane by using Euclidean matrix multiplication In our former study [, by introducing Lorentz matrix multiplication, special matrices and the motions on Lorentz plane were defined without the need of sign matrix By this way, convenience in between the matrix multiplication and scalar product in Lorentzian was obtained Further definition of the matrix multiplication using scalar product R n of which index is ν were also given in [ Our aim in the present paper is to define a new matrix multiplication being compatible with any scalar product in R n Some applications were also given Let R m,n be the set ofall m n matrices R m,n with the matrix addition and the scalar-matrix multiplication is a real vector space More properties of the ordinary matrix multiplication can be found in [4 Let g be a scalar product on R n which is nondegenerate symmetric bilinear form g(x,y) = g ij x i y j, i,j= Received September, 7; accepted October, 7 Mathematics Subject Classification 5A8 789

2 79 O Keçilioğlu and H Gündoğan where x = (x,x n ),y = (y,,y n ) R n For the standard basis {e i } i n of R n, the matrix representation of scalar product is G = [g ij = g(e i,e j ) R n,n Generalized Matrix Multiplication and Properties Let A,,A m denote the row vectors of A = [a ij R m,n and B,,B p denote the column vectors of B = [b jk R n,p Then we define a new matrix multiplication denoted by g, as A g B = g ( A,B ) g ( A,B ) g(a,b p ) g ( A,B ) g ( A,B ) g(a,b p ) g ( A n,b ) g ( A n,b ) g(a n,b p ) = [ g ( A i,b j) We call this multiplication as generalized matrix multiplication and if we let A i to be ith rowofaand B j tobe jth columnofb then (i,j) entryofa g B isg ( A i,b j) Note that A g B is an m p matrix We will denote R m,n with generalized matrix multiplication by R m,n g In the special case of g we get followings: For g(x,y) = n x i y i, A g B coincides with usual matrix multiplication i= i= For g(x,y) = x y + n x i y i, A g B coincides with Lorentzian matrix multiplication defined in [ For g(x,y) = ν x i y i + n i= multiplication defined in [ i=ν+ Theorem The following statements are satisfied x i y i, A g B coincide with Pseudo matrix For every A R m,n g,b R n,p g,c R p,r h, A g (B h C) = (A g B) h C For every A R m,n g,b,c R n,p g, A g (B +C) = A g B +A g C For every A,B R m,n g,c R n,p g, (A+B) g C = A g C +B g C 4 For every k R, A R m,n g,b R n,p g, k(a g B) = (ka) g B = A g (kb) where g and h are nondegenerate symmetric bilinear forms on R n and R p, respectively

3 Generalized Matrix Multiplication 79 Definition The determinantofamatrix A = [a ij R n,n g and defined as det A = s(σ)a σ() a σ() a σ(n)n, σ S n is denoted by deta where S n is set of all permutations of the set {,,,n} and s(σ) is sign of the permutation σ Let {E ij } be the standard basis of R n,n and X = [x ij R n,n Consider the linear equation system X g E ij = E ij for i,j n If we handle the equations having the variables {x,x,,x n }, we obviously get g i x i =, i= g ij x ji =, ( j n) i= Denoting the first column matrix of X by X, the ordinary matrix form of above system can be stated as G T X = Since G is symmetric and regular matrix, we deduce X = G Ifweproceedthesamewayfortheequationshavingthevariables{x j,x j,,x jn }, j n, we have X = [ g ij, detg where g ij is (i,j) cofactor of G Now we are ready to give the definition of identity matrix Definition Let g be a nondegenerate symmetric bilinear form on R n, then n n identity matrix according to generalized matrix multiplication, is denoted by I n and defined by I n = [ g ij detg

4 79 O Keçilioğlu and H Gündoğan Note that for every A = [a ij R n,n ν, I n g A = A g I n = A Indeed, Corollary R n,n g I n g A = I n g = = a ij E ij i,j= a ij (I n g E ij ) i,j= a ij E ij i,j= = A with generalized matrix multiplication is an algebra with unit Definition An n n matrix A is g-invertible, if there exists an n n matrix B such that A g B = B g A = I n Then B is called g-inverse of A and is shown by A g Example [ Let g(x,y) = x y [ +x y +x y +x y be [ a scalar product on R and 4 A = R Then I = and A 9 g = 7 Definition 4 Let A = [a ij be an m n matrix Then the transpose of A is the n m matrix A T obtained by interchanging the rows and columns of A, so that the (i,j)th entry of A T is a ji Theorem Let A and B be matrices of the appropriate sizes so that the following operations make sense, and c be a scalar Then (A+B) T = A T +B T (A g B) T = B T g A T (ca) T = ca T 4 ( A T) T = A Proof We only prove the property () The others can be easily done (A g B) T = [ g ( A i,b j) T 4 7 = [ g ( A j,b i) = [ g ( B i ),A j [ ( (B T = g ),( A T) ) j i = B T g A T

5 Generalized Matrix Multiplication 79 Definition 5 Let A R n,n g If A T = A and A T = A then A is said to be symmetric and skew-symmetric matrix, respectively Definition 6 Let A be an m n matrix and denote the rows of A by A i, i =,,m The elementary row operations on matrix A are: R ij, the interchange of rows A i and A j, R i (c), the multiplication of row A i by the nonzero scalar c, R ij (a), the addition of a times row A j to row A i, i j Definition 7 Let I n be identity matrix according to generalized matrix multiplicationobtainedfromg nondegeneratesymmetricbilinearformonr n Anelementary matrix is an n n matrix obtained from the identity matrix I n by performing on I n a single elementary row operation For the operations R ij,r i (c) and R ij (a), elementary matrices denoted by E ij,e i (c) and E ij (a), respectively Let I n be identity matrix according to generalized matrix multiplication obtained from g(x,y) = x y +x y x y x y x y nondegenerate symmetric bilinear form on R Then I = and the following are examples of elementary matrices: E =,E ( 6),E () Theorem Let R be an elementary row operation and let E be the corresponding m m elementary matrix Then, where A is any m n matrix R(A) = E g A Proof The proof is similar to standard one Theorem 4 The determinants of the elementary matrices are given as dete ij = deti n dete i (c) = cdeti n

6 794 O Keçilioğlu and H Gündoğan dete ij (a) = deti n Proof We omit the details since it is clear from the properties of determinant function Theorem 5 Let E be an elementary matrix and A be an n n matrix Then the following holds det(e g A) = (deti n ) dete deta Proof Let A be a n n matrix and E be an elementary matrix Firstly, by taking E = E i (c), we get det(e f A) = det(e i (c) f A) The properties of determinant function, Theorem and Theorem 4 leads to det(e i (c) g A) = c deta = (deti n ) dete i (c) deta For the other cases E = E ij (a) and E = E ij, the proof can be done similarly Theorem 6 For every A,B R n,n g, det(a g B) = (deti n ) deta detb Proof Let A be a matrix obtained by multiplying elementary matrices E and E From above theorem, we get det(a g B) = det((e g E ) g B) = det(e g (E g B)) = (deti n ) dete det(e g B) ( = (deti n ) ) dete dete detb = (deti n ) det(e g E )detb = (deti n ) detadetb For A = E g g E k, where E i is elementary matrix ( i k), the proof can be done similarly On the other hand, since every regular matrix can be written as the multiplication of elementary matrices, for every regular matrix A det(a g B) = (deti n ) deta detb If A is a singular matrix, then deta = and det(a g B) = Thus the proof is completed Corollary If A is a regular matrix then deta Eigenvalues and eigenvectors play an important role in matrix theory because of its application in the areas of mathematics, physics and engineering By this aim, we define the eigenvalues and eigenvectors of square n n matrix A by generalized matrix multiplication

7 Generalized Matrix Multiplication 795 Definition 8 Let A R n,n g An eigenvector of A is a nonzero vector x in R n such that A g x = λx for some scalar λ The scalar λ is called an eigenvalue of the matrix A, and we say that the vector x is an eigenvector belonging to the eigenvalue λ Example Let g(x,y) = x y +x y + x y + x y + x y be a scalar product on R and A = R, g Theneigenvalues of Aare λ =,λ = andλ = 5 Someeigenvectors ofacorresponding to λ,λ and λ are respectively u = 6 5 4, u =, u = Theorem 7 The eigenvectors of a symmetric matrix A R n,n g to different eigenvalues are orthogonal to each other corresponding Proof For the eigenvectors x, y corresponding to two different eigenvalues λ, µ of the matrix A, we can say that A g x = λx and A g y = µy, so () y T g A g x = λy T g x = λg(x,y) But numbers are always their own transpose, so () y T g A g x = x T g A g y = x T g µy = µ ( x T g y ) y T g A g x = µg(x,y) From () and (), we get (λ µ)g(x,y) = So λ = µ or g(x,y) =, and it isn t the former, so x and y are orthogonal Example Let g(x,y) = x y +x y +x y +x y +x y and A = R, g

8 796 O Keçilioğlu and H Gündoğan A is a symmetric matrix Then the eigenvalues of A are λ = 5+7,λ = 7 5 and λ = Some eigenvectors of A corresponding to λ,λ and λ are u = respectively For i j, we get 5 5, u = 5 5, u = g(u i,u j) = u T i g u j = Then {u,u,u } form an orthogonal basis of R g Throughout the rest of the paper, g will denote the scalar multiplication function g(x,y) = n g i x i y i, for the convenience of readers, where g i for i n i= Hence, the unit matrix corresponding to the generalized matrix can be stated as I n = g g g n Definition 9 Let A R n,n g If A = A T ( ieaa T = A T A = I n ) then A is said to be g-orthogonal matrix Theorem 8 Let A R n,n g Then A is g-orthogonal if and only if the row vectors of A form an orthogonal basis of R n g under the scalar product; and A is g-orthogonal if and only if the column vectors of A form an orthogonal basis of R n g under the scalar product Proof We will only prove (), since the proof of () is almost identical Let A,,A n denote the row vectors of A Then A g A T = g(a,a ) g(a,a n ) g(a n,a ) g(a n,a n ) It follows that A g A T = I n if and only if for every i,j =,,n 5 A i,a j = { g i, i = j, i j Then {A,,A n } is an orthogonal basis of R n g

9 Generalized Matrix Multiplication 797 Definition A matrix A is diagonalizable if there exists a nonsingular matrix P and a diagonal matrix D such that D = P g A g P Theorem 9 Let all the eigenvalues of A R n,n g are real Then A is diagonalizable if and only if it has n linearly independent eigenvectors Proof Let x,,x n be n linearly independent eigenvectors of A associated with the eigenvalues λ,,λ n That is, Now, we denote P = [ x A g x i = λ i x i, i =,,n x n Since the columns of P are linearly independent, P is invertible Let D be diag[ λ g,, λn g n Then A g P = A g [ x x n = [ λ x λ n x n = [ x x n g diag = P g D [ λ,, λ n g g n Since A g P = P g D, it follows that D = P g A g P which shows that A is diagonalizable To prove the other direction we assume that A is diagonalizable [ Then there λ exists a nonsingular matrix P and a diagonal matrix D = diag g,, λn g n such that D = P g A g P If we multiply above equation with P from the left, we get which implies () A g v i = λ i v i A g P = P g D, i =,,n where v i are columns of P The equations () show that v,,v n areeigenvectors of A corresponding to eigenvalues t,,t n Furthermore, since P is invertible, {v,,v n } are linearly independent Example 4 Let g(x,y) = x y x y +x y and A = R g

10 798 O Keçilioğlu and H Gündoğan The eigenvalues of A are λ =,λ =,λ = and eigenvectors corresponding these eigenvalues are u =, u =, u = respectively Therefore P = and P = Finally P g A g P = REFERENCES A A Ergin: On the -parameter Lorentzian motions Comm Fac Sci Univ Ankara Ser A Math Statist 4 (99), H Gündoğan and O Keçilioğlu: Lorentzian matrix multiplication and the motions on Lorentzian plane Glas Mat Ser III 4(6), no (6), 9 4 H Gündoğan and O Keçilioğlu: Pseudo Matrix Multiplication Commun Fac Sci Univ Ank Sér A Math Stat 66, no, (7), S Lang: Linear Algebra Addison-Wesley Publishing Co, London, 97 5 B O Neill: Semi-Riemannian Geometry With Applications to Relativity Academic Press, New York, 98 Osman Keçilioğlu Faculty of Science and Arts Department of Mathematics Krkkale University 745 Krkkale, Turkey okecilioglu@yahoocom Halit Gündoğan Faculty of Science and Arts Department of Mathematics Krkkale University 745 Krkkale, Turkey hagundogan@hotmailcom

LORENTZIAN MATRIX MULTIPLICATION AND THE MOTIONS ON LORENTZIAN PLANE. Kırıkkale University, Turkey

GLASNIK MATEMATIČKI Vol. 41(61)(2006), 329 334 LORENTZIAN MATRIX MULTIPLICATION AND THE MOTIONS ON LORENTZIAN PLANE Halı t Gündoğan and Osman Keçı lı oğlu Kırıkkale University, Turkey Abstract. In this