KRONECKER PRODUCT AND LINEAR MATRIX EQUATIONS

Transcription

1 Proceedings of the Second International Conference on Nonlinear Systems (Bulletin of the Marathwada Mathematical Society Vol 8, No 2, December 27, Pages 78 9) KRONECKER PRODUCT AND LINEAR MATRIX EQUATIONS R B Shenvi Khandeparkar, Department of Mathematics, Smt Parvatibai Chowgule College of Arts and Science, MARGAO GOA, ramshila2@yahoocom Abstract This article discusses Kronecker Product and its properties Properties are verified with illustrations As an application of Kronecker Product the second part of the article deals with Linear Matrix Equations We also discuss the solution of Linear Matrix Equations using Kronecker Product and its properties Attempt is made to provide one practical illustration which is being solved using a property of the Kronecker Product 1 INTRODUCTION Kronecker Product is also known as a direct product or a tensor product The concept of Kronecker product has its origin in theory of groups and has been found useful in particle physics It extended the scope of solving linear system of equations,and linear system of differential equations The Kronecker product has been profitably employed in the study of circulant matrices The developments in the Kronecker product and matrix calculus lead readers to the frontiers of modern developments in matrix theory and its applications in many real life situations 2 THE VEC OPERATOR A vector valued function of a matrix A denoted by vec A is defined as follows Let A be any matrix of order m n then vec A A 1 A 2 A n 78

2 Kronecker Product and Linear Matrix Equations 79 Observe that A j is a column vector of order m Hence, vec A represents a vector of the order mn [ ] a11 a For the matrix A 12 a 13 a 21 a 22 a 23, 2 3 vec A [a 11 a 12 a 13 a 21 a 22 a 23 ] T [ ] 2 3 For a Matrix A, veca [ ] T KRONECKER PRODUCT Let A [a ij ] m n and B [b ij ] p q be two arbitrary matrices The Kronecker product of A and B, denoted as A B, is defined as the partitioned matrix of order (mp nq) given by a 11 B a 12 B a 1n B A B a 21 B a 22 B a 2n B a m1 B a m2 B a mn B (mp nq) It is a general concept of matrix multiplication irrespective of the order of two matrices Any (i, j) th element of A B is a matrix [a ij B] of order p q For matrix of order two [ ] [ ] a11 a let A 12 b11 b, B 12 a 21 a 22 b 21 b 22 Then [ ] a 11 b 11 a 11 b 12 a 12 b 11 a 12 b 12 a11 B a A B 12 B, a 11 b 21 a 11 b 22 a 12 b 21 a 12 b 22 a 21 B a 22 B a 21 b 11 a 21 b 12 a 22 b 11 a 22 b 12 a 21 b 21 a 22 b 22 a 22 b 12 a 22 b 22 Next, consider the linear transformations x Ay and z Bw defined by, [ ] [ ] [ ] [ ] [ ] [ ] x1 a11 a 12 y1 z1 b11 b and 12 w1 x 2 a 21 a 22 y 2 z 2 b 21 b 22 w 2 These two transformations are coupled simultaneously by defining two vectors ν given by x 1 z 1 y 1 w 1 µ x z x 1 z 2 x 2 z 1, and ν y w y 1 w 2 y 2 w 1 x 2 z 2 y 2 w 2 µ and

3 8 R B Shenvi Khandeparkar Now, the component x 1 z 1 of µ is given by Similarly, other components are evaluated x 1 z 1 (a 11 y 1 + a 12 y 2 )(b 11 w 1 + b 12 w 2 ) (a 11 b 11 )(y 1 w 1 ) + (a 11 b 12 )(y 2 w 2 ) This evaluation leads to the transformation a 11 b 11 a 11 b 12 a 12 b 11 a 12 b 12 µ a 11 b 21 a 11 b 22 a 12 b 21 a 12 b 22 a 21 b 11 a 21 b 12 a 22 b 11 a 22 b 12 ν, a 21 b 21 a 21 b 22 a 22 b 21 a 22 b 22 i e µ (A B)ν x z (A B)(y w), i e Ay Bw (A B)(y w) 4 PROPERTIES OF KRONECKER PRODUCT We list below important properties of Kronecker product without proof 1 A (αb) α(a B), where α is a scalar ; 2 (A + B) C (A C) + (B C) ; 3 A (B + C) (A B) + (A C) ; 4 A (B C) (A B) C ; 5 (A B)(C D) AC BD ; 6 A B Ā B ; 7 (A B) T A T B T ; (A B) A B ; ( * denotes conjugate transpose ) 8 ρ(a B) ρ(a)ρ(b); ρ(a) denotes rank of A In all the above mentioned properties, matrices A, B and C are rectangular matrices and that the operations are assumed to be defined For additional two more properties, assume that A and B are square matrices of orders m and n respectively Then we have, 9 (A B) 1 A 1 B 1 (A, B and A B are non - singular matrices ) 1 det (A B) (deta) n (detb) m Complete proofs of each of the above properties is given in [ 1 ]

4 Kronecker Product and Linear Matrix Equations 81 5 VECTOR VALUED FUNCTION The following useful relation is true for matrices A(m n), Y (n p) and relation is Vec(AY B) (B T A) vec Y B(p s) The The proof depends on comparing the elements on either sides The following illustrations verify the above property for matrices of order 2 2 Illustration 1 : Let A [a ij ], B [b ij ] and Y [y ij ] be matrices all of order (2 2), verify that We have, Vec(AY B) (B T A) vecy a 11 y 11 b 11 + a 12 y 21 b 11 + a 11 y 12 b 21 + a 12 y 22 b 21 vec A Y B a 21 y 11 b 11 + a 22 y 21 b 11 + a 21 y 12 b 21 + a 22 y 22 b 21 a 11 y 11 b 12 + a 12 y 21 b 12 + a 11 y 12 b 22 + a 12 y 22 b 22, a 21 y 11 b 12 + a 22 y 21 b 12 + a 21 y 12 b 22 + a 22 y 2 b 22 [ ] [ ] b 11 a 11 b 11 a 12 b 21 a 11 b 21 a 12 B T b11 b A 21 a11 a 12 b 11 a 21 b 11 a 22 b 21 a 21 b 21 a 22 b 12 b 22 a 21 a 22 b 12 a 11 b 12 a 12 b 22 a 11 b 22 a 12, b 12 a 21 b 12 a 22 b 22 a 21 b 22 a 22 y 11 with vec Y y 21 y 12, [BT A] vec Y vec A Y B y 22 Illustration 2 : Consider the equation [ ] [ ] [ ] a11 a 12 y1 y 3 b11 b 12 a 21 a 22 y 2 y 4 b 21 b 22 We can write the equation as AY I 2 B Observe that, vec (AY I 2 ) (I 2 A) vecy vecb, which is a vector matrix form of the equation given by a 11 a 12 y 1 b 11 a 21 a 22 a 11 a 12 a 21 a 22 y 2 y 3 y 4 b 21 b 12 b 22

5 82 R B Shenvi Khandeparkar 6 EIGENVALUES AND EIGENVECTORS If {λ i } and {x i } are the eigenvalues and the corresponding eigenvectors of a matrix A of order n n ; and {µ i } and {y i } are the eigenvalues and the corresponding eigenvectors of a matrix B of order m m, then the Kronecker product (A B) has eigenvalues {λ i µ i } with the corresponding eigenvectors {x i y i } This property is important since it extends the scope of finding eigenvalues and eigenvectors of a class of matrices of higher orders Further, this information is useful in finding eigenvalues of the matrices f(a B) where f(x) is a polynomial expression in x [ ] [ ] Illustration 3 : Consider A and B The characteristic equation for matrix A is A λi 2 It can be verified that the eigenvalues of A are 1 and 3 and corresponding eigenvectors are respectively x 1 [1, 1] T and x 1 [1, 1] T The characteristic equation for matrix B is B µi 2 It can be verified that the eigenvalues of B are 1 and 1 and corresponding eigenvector is y 1 [1, ] T Next, we have A B [ ] [ ] The characteristic equation for the matrix A B is A B αi 4 1 α ie 1 α α α This gives (1 α) 4 16 By inspection α 1 or α 3 The eigenvalues of A B are 1 and 3 Corresponding to the eigenvalue 1, the associated eigenvectors are given by the linear system 2x 2y + 2z 4w, 2y + 2w, 2x 4y + 2z 2w, 2y + 2w ; and any eigenvector associated with the eigenvalue 1 is given by [1,, 1, ] T

6 Kronecker Product and Linear Matrix Equations 83 Next, corresponding to the eigenvalue 3, the associated eigenvectors are given by the linear system 2x 2y + 2z 4w, 2y + 2w, 2x 4y 2z 2w, 2y 2w ; and any eigenvector associated with the eigenvalue 3 is given by [1,, 1, ] T Thus, for the matrix A, λ 1 1, x 1 [1, 1] T and λ 2 3, x 2 [1, 1] T, for the matrix B, µ 1 1, y 1 [1, ] T and note that µ 2 µ 1, y 2 y 1, For matrix A B, α 1 λ 1 µ 1 1 and α 2 λ 2 µ 2 3 The corresponding eigenvectors are x 1 y 1 [1,, 1, ] T and x 2 y 2 [1,, 1, ] T 7 PROPERTY OF THE KRONECKER SUM Given a matrix A of order m m and a matrix B of order n n, the Kronecker sum of A and B, denoted by A B, is defined as the expression A B A I n + I m B If {λ i } and {µ j } are the eigenvalues of A and B respectively, then {λ i + µ j } are the eigenvalues of A B We verify this property with the following illustration Illustration 4 : Verify the above result for A [ ] and B [ ] Observe that the eigenvalues of A are λ 1 1 and λ 2 3 and the corresponding eigenvectors are respectively x 1 [1, 1] T and x 2 [1, 1] T The eigenvalues of B are µ 1 1 and µ 2 2 and the corresponding eigenvectors are respectively y 1 [1, ] T and y 2 [1, 1] T Next observe that, α λ 1 + µ 1 and x 1 y 1 [1,, 1, ] T, α 1 λ 1 + µ 2 and x 1 y 2 [1, 1, 1, 1] T, α 4 λ 2 + µ 1 and x 2 y 1 [1,, 1, ] T, α 5 λ 2 + µ 2 and x 2 y 2 [1, 1, 1, 1] T

7 84 R B Shenvi Khandeparkar 8 LINEAR MATRIX EQUATIONS Theory of matrices is one of the important mathematical instruments to resolve many complex situations In particular, these methods have been widely employed in the analysis of systems of equations Matrix methods are also useful in several problems of modeling in engineering, technological sciences, social sciences With the use of computers, large scale systems can be solved by enhancing the scope of applications The vector matrix equation is then generalized to solve the systems of linear equations AX B, AX + XB C, AXB C Kronecker product methods and properties are useful in matrix functional equations of this type 9 MORE GENERAL LINEAR MATRIX EQUATIONS In the last two decades many more new linear matrix equations have been studied with several details bringing out applications to new areas of science and technology Some of these are given below Interested readers may explore the reference for more information Consider a linear matrix equation AX + XB C, (91) where A, B, C are given matrices of order n, and X of order n is an unknown matrix These are called Lyapunov equations The aim is to find a unique matrix X which verifies (91) The equation (91) is fairly general and includes the following particular cases 1 AX C when B O, 2 AX ± XA O, when B A and C, 3 AX ± XA T, 4 AX ± XA 1 It is claimed that X e At Ce Bt dt (92) satisfies the equation provided the integral exists for arbitrary matrix C of order n Let us verify this claim Clearly, the relation ( 92) is well defined Further AX BX Ae At Ce Bt dt e At Ce Bt Bdt d ( e At ) Ce Bt dt, and dt ( e Bt ) dt e At C d dt

8 Kronecker Product and Linear Matrix Equations 85 Hence, d ( AX + XB e At Ce Bt) dt [ (e At Ce Bt] dt C The following conditions need to be satisfied by the matrices A and B so that the equation ( 92 ) has a solution Let λ i (i 1, 2,, n) be the eigenvalues of A and µ j (j 1, 2,, n) be the eigenvalues of B The necessary and sufficient condition that ( 92 ) has a solution for all C is that (λ i + µ j ) The Proof is omitted There is an alternative method to solve ( 91 ) Here some of the properties of Kronecker product come to help These are (i) Vec(AXI n ) (I n A)VecX, (I n Identity matrix), (ii) Vec(I n XB) (B T I n ))VecX Now AX + XB AXI n + I n XB HenceVec (AX + XB) Vec[AXI n + I n XB] Vec[AXI n ] + Vec[I n XB] [(I n A) + (B T I n )]VecX (B T A)VecX VecC (93) Thus the Kronecker product properties convert the given equation into matrix vector form This equation has a unique solution if and only if (B T A) is non - singular, which implies that the eigenvalues of (B T A) are all non-zero If the eigenvalues of A are λ i and the eigenvalues of B are µ j, then the eigenvalues of (B T A) are (λ i +µ j ) Therefore the equation ( 91 ) has a unique solution if and only if (λ i + µ j ) which means that matrices A and B have no eigenvalues in common To solve the Lyapunov s matrix equation, when matrices A and B are given, find (i)(b T A), (ii)vecx and (iii)vecc Then we get the matrix vector equation (9 3 ) After solving this equation convert the solution Vec X into a matrix X The following illustration explains the procedure Illustration 5 : Obtain the solution to the linear matrix equation ( 91 ) where A [ ] 2 2, B 2 5 [ ] 2 2 and C [ ] To solve ( 91 ) we now follow the procedure given above From ( 93 ), we get (B T A) [(I 2 A) + (B T I 2 )]

9 86 R B Shenvi Khandeparkar Hence, we obtain a matrix vector equation x x x x 4 5 having the solution Vec X [ ] T Reconverting the vector into a matrix, we have [ ] [ ] x1 x X x 3 x Consider the matrix equation of the form AX XA µx (94) where A and X are n n matrices We have, AX AXI n gives Vec (AXI n ) (I n A) Vec X, XA I n XA gives Vec(I n XA) (A T I n ) Vec X Let (I n A) (A T I n ) H, then equation ( 94 ) takes the matrix vector form H VecX µ Vec X (95) This equation has non - trivial solution for Vec X if and only if µi H or µ is an eigenvalue of H Here I n is an identity matrix of order n But the eigenvalues of H are (λ i λ j ), where {λ i }i 1, 2,, n are eigenvalues of A Hence equation (94) has non - trivial solution if and only if µ λ i λ j, where i, j 1, 2,, n, i j [ ] 5 3 Illustration 6 : Let A in (9 4) 1 1 The eigenvalues of A are λ 1 2 and λ 2 4 We have, µ 2 is an eigenvalue of H Equation ( 95 ) gives x 1 y x y We get the system of equations, z w 2 2x + 3y + z, x 2y + w, 3x + 6z + 3w, 3y z + 2w z w

10 Kronecker Product and Linear Matrix Equations 87 On solving we obtain x 1, y 1, z 1, w 1 Then we have [ ] 1 1 X 1 1 Illustration 7 : A medical expert advised two patients Usha and Nisha of different ages to consume (15, 145) gms of protein and (33, 27) gms of carbohydrates respectively The patients consume two types of food and when mixed, provide them approximately the desired food values Noticing that in course of time food values of P and Q decrease, the expert advised them to add vitamins R and S containing proteins and carbohydrates as boosters The following tables present food analysis of of P and Q and vitamins R and S Type of food Food value 1 gms Proteins Carbohydrates P 5 15 Q 3 25 The food items are boosted by vitamins R and S having potency as follows Type of vitamins Food value 1 gms Proteins Carbohydrates R 5 2 S 1 5 What amount of items P and Q should Usha and Nisha consume so that they receive the advised food values? Let x 11 and x 21 be the quantities in units of 1 gms of P and Q for Usha and let x 12 and x 22 the quantities for Nisha These quantities are then subjected to vitamins R and S The resulting model is then given by a matrix equation [ ] [ x11 x 12 x 21 x 22 ] + [ ] [ ] x11 x x 21 x [ ] , ie AX + XB C Now we have (B T A) [(B T I 2 ) + (I 2 A)] [ ] [ ] [ ] 1 1 [ 5 ] Using equation (9 3 ), we get x x x x 22 27

11 88 R B Shenvi Khandeparkar We get the system of equations, 1x x x 12 15, 3x x x 22 33, 2x x x , 2x x x Solving, we get x 11 4, x 21 6, x 12 2, x 22 3 Thus, Usha consumes 4 units and 6 units and Nisha consumes 2 units and 3 units of food P and q respectively Yet another kind of linear matrix functional equation which occurs in literature has the form AXB C (96) where A, B, C are given square matrices of order n n and X is an unknown matrix of the same order Here the matrices A and B do not commute This equation becomes AX C when B I n We have, from the property of Kronecker Product, ( see article 4), Vec(AXB) (B T A) VecX VecC (97) iehx c(x VecX, c VecC) (98) ( 98 ) is a matrix vector equation of the form (94) Clearly if det H, the solution is x H 1 c Illustration 8 : In equation ( 96 ), let A [ ], B [ ] The linear matrix equation ( 96 ) becomes [ ] [ ] [ ] 4 3 x1 x x 2 x [ ] 8x3 + 6x ie 4 12x 1 + 9x 2 4x 3 3x 4 2x 3 + 4x 4 3x 1 + 6x 2 x 3 2x 4 Equating the corresponding elements on two sides, we get [ ] [ ] x 3 + 6x 4 2, 12x 1 + 9x 2 4x 3 3x 4 4, 2x 3 + 4x 4 2, 3x 1 + 6x 2 x 3 2x 4 3 Solving, x 1 1, x 2 1, x 3 1, x 4 1, ie X [ ]

12 Kronecker Product and Linear Matrix Equations 89 Using the Kronecker product method, we have [ ] [ ] H B T A In matrix vector form the equation ( 98 ) becomes 8 6 x [ ] x1 x X 3 x 2 x x 2 x 3 x [ ] 1 1 verifies the given equation 1 1 Consider the more general form of system equation ( 96 ) where A 1, A 2, B 1, B 2 are matrices of order n, given below A 1 XB 1 + A 2 XB 2 C (99) We have Vec(A 1 XB 1 ) (B T 1 A 1 )VecX, and Vec(A 2 XB 2 ) (B T 2 A 2 ) Vec X Let H (B T 1 A 1) + (B T 2 A 2) The system ( 99 ) reduces to the form Hx c where x Vec X, c Vec C We have, Illustration 9 : Find the matrix X in (99) given that [ ] [ ] [ ] [ ] A 1, B 1 2 1, A 1 2, B B T 1 A , BT 2 A ; and Hx c given by the matrix vector equation 5 2 x x x x 4 4 This system when solved leads to the solution of ( 99), namely [ ] [ ] x1 x X x 2 x and C [ ] 3 3 4

13 9 R B Shenvi Khandeparkar Acknowledgement I thank Dr S G Deo, ex Head of the Department of Mathematics, and Dean, Goa University, for support and help in preparing the article References [1] Graham Alexander,Kronecker Products and Matrix Calculus with Applications, John Wiley & Sons, NY,(1981) [2] Herstein IN,David J Winter,Matrix Theory and Linear Algebra, Macmillan, New York,(1989)