On Hadamard and Kronecker Products Over Matrix of Matrices

Transcription

1 General Letters in Mathematics Vol 4, No 1, Feb 2018, pp13-22 e-issn , p-issn Available online at wwwrefaadcom On Hadamard and Kronecker Products Over Matrix of Matrices Z Kishka1, M Saleem 2, M Abdalla3 and A Elrawy 4 1,2 Dept of Math, Faculty of Science, Sohag University, Sohag 82524, Egypt Dept of Math, Faculty of Science, South Valley University, Qena 83523, Egypt 1 zanhomkiska@yahoocom, 2 abuelhassan@yahoocom,3 mabdallah@scisvuedueg,4 amrmath1985@yahoocom, 3,4 Abstract In this paper, we define and study Hadmard and Kronecker products over matrix of matrices and their basic properties Furthermore, we establish a connection the Hadamard product of matrix of matrices and the usual matrix of matrices multiplication In addition, we show some application of the Kronecker product Keywords: Hadamard Schur product, Kronecker sum, Kronecker product, matrix of matrices 2010 MSC No: 15A15, 15A09, 34A30, 39A10 1 Introduction Matrices and matrix operations play an important role in almost every branch of mathematics, computer graphics, communication, computational mathematics, natural and social sciences and engineering The Hadamard and Kronecker products are studied and utilized widely in matrix theory, statistics [1, 2], physics[2], system theory and other areas; see, eg [3, 4, 5, 6, 7, 8, 9, 10, 11] An equality connection between the Hadamard and Kronecker products look to be firstly used by eg [12, 13, 14] In partitioned matrices, the Khatri-Rao product can be seen as a generalized Hadamard product which is discussed and used by many authors eg [15, 16] Also Tracy-Singh product as a generalized Kronecker product is studied in [19, 20, 21] Finally, the approach of this paper may not be practical conventional in all situations In the present paper, we define and study Hadamard and Kronecker product over the matrix of matrices in a short way; MMs which was presented newly by Kishka et al [22] We now give a short overview of this paper In section 2, we define and study Hadamard product over MMs and give some properties Then we show the association between Hadamard and MMs product in section 3 Finally, in section 4, we introduce the Kronecker product and prove a number of its properties In addition, we introduce the notation of the vector matrices VMs-operator from which applications can be submitted to Kronecker product Throughout this paper, the accompanying notations are utilized: Let K be a field and Ml K be the set of all l l matrices defined on K, I and O stand for the identity matrix and the zero matrix in Ml K, respectively We denote by Mm n Ml K the set of all m n MMs over Ml K, the elements of Mm n Ml K are denoted by A, B, C, D, G, [22] Let A Mm n Ml K, then it can be written as a rectangular table of elements Aij ; i 1, 2,, m and j 1, 2,, n, as follows: A11 A1n A Am1 Amn

2 14 ZKishka et al Definition 11 [22] Given a m n-mms; A A ij Its transpose is the n m-mms A t, given by where A ij M l K [A t ] ji A ij [A] ij, i 1,, m; j 1,, n, 2 Hadamard Product In this section, we give a definition of Hadamard product over MMs with some properties Definition 21 Let A A ij and B B ij M m n M l K, then A B A ij B ij, i 1,, m, j 1,, n, where A ij, B ij M l K, this product is called Hadamard or Schur product over MMs Example 22 Let A and B M 2 M 2 K where A , 2 4 and then 1 1 B , 1 2 A B Right away we might investigate some fundamentals properties of the Hadamard product over MMs Theorem 23 Let A and B M m n M l K, then A B B A if M l K is commutative ring Proof The proof follows directly from the fact that A ij and B ij are commutative Let A and B be m n-mms where A A ij and B B ij, then A B A ij B ij B ij A ij B A Theorem 24 The identity MMs under the Hadamard product is the m n MMs with all entries equal to I, denoted J mn Proof Let A A ij M m n M l K, then A J mn A ij I ij A ij and so J mn A A ij Therefore, J mn as defined above is indeed the identity MMs under the Hadamard product We have denoted the Hadamard identity as J mn to avoid confusion with the usual identity MMs, I n Theorem 25 Let A A ij M m n M l K, then A has a Hadamard inverse, denoted by A ˆ if and only if A ij are non-singular i 1, 2,, m; j 1, 2,, n Furthermore, A ˆ A 1 ij

3 On Hadamard and Kronecker products over Matrix of Matrices 15 Proof Let A A ij M m n M l K with Hadamard inverse A, ˆ then A A ˆ A ij A 1 ij Iij J mn, which is only possible when all entries of A are invertible In other words A ij are non-singular matrices i 1, 2,, m; j 1, 2,, n Take any A A ij M m n M l K such that A ij are non-singular i 1, 2,, m; j 1, 2,, n Then there exists A ˆ A 1 ˆ such that A A A ˆ A J mn, and so A has an inverse A ˆ A 1 ij We have denoted the Hadamard inverse as ˆ A to avoid confusion with the usual inverse MMs, A 1 Theorem 26 Let A, B M m n M l K, then A B t A t B t Proof Suppose that A, B M m n M l K where A A ij and B B ij, then A B t A ij B ij t A ji B ji A t B t ij Remark 27 If M l K is commutative ring with unity, then A B t B t A t Theorem 28 The set M n M l K of non-singular MMs is a group under Hadamard product Proof Let A A ij, B B ij and C C ij M n M l K be MMs then it is easy to see that M n M l K is a group Theorem 29 Suppose that E M l K and A, B and C M n M l K, then 1 C A + B C A + C B 2 E A B EA B A EB where M l K is commutative ring with unity Proof Part 1 Part 2 [C A + B] ij [C] ij [A + B] ij [C] ij [A] ij + [B] ij [C] ij [A] ij + [C] ij [B] ij [C A] ij + [C B] ij [C A + C B] ij since M l K is commutative ring with unity [E A B] ij E[A] ij [B] ij [EA] ij [B] ij [EA B] ij [A] ij E[B] ij [A] ij [EB] ij [A EB] ij, Remark 210 The above part 2 is also true if we replace E by c K

4 16 ZKishka et al 3 Connection between Hadamard and MMs products In this section, we show connection between Hadamard and MMs multiplications Let A A ij M m M l K and consider the set S 12m of m cyclic permutations of 12m given by S 12m {123 m 1 m, 23 m 1 m1,, m12 m 2 m 1} 1 If A i is the i th column of the m m MMs A, then we define A s A s1 : A s2 : : A sm, 2 ie, A s is the MMs A with permuted columns according to the permutation s s 1 s 2 s m It is easy to see that A s AI s Also, for B B ij M m M l K, let us define B s11 B s22 B smm B s11 B s22 B smm B [S] 3 B s11 B s22 B smm With these notation, we have the following theorem Theorem 31 Let A A ij and B B ij M m M l K, then AB s A s B [s] S AIs B[s], where A S and B [S] are defined as in 2 and 3, respectively, for a particular permutation s S 12m Proof Suppose that A A ij and B B ij M m M l K, then m j1 A m 1jB j1 j1 A 1jB jm m j1 AB A m 2jB j1 j1 A 2jB jm m j1 A m mjb j1 j1 A mjb jm m m m A ij B j1 : A ij B j2 : : A ij B jm j1 j1 Clearly, the usual product of MMs AB decomposes uniquely as the sum of m MMs C s, where s s 1 s 2 s m S 123m Such a decomposition can be constructed as follows: The first MMs C 12m is obtained by C 12m A i1 B 11 : A i2 B 22 : : A im B mm A 12m B [12m] The second MMs is selected according to the permutation 23m1, ie, C 23m1 A i2 B 21 : A i3 B 32 : : A i1 B 1m A 23m1 B [23m1] Following this procedure and taking the MMs according to the complete set of m cyclic permutations of 12m in 1, we obtain the m 1 th MMs as C m 1m1m 3m 2 A m 1m1m 3m 2 B [m 1m1m 3m 2] Finally, the MMs corresponding to the ultimate permutation m1 m 2m 1 is formed by the remaining summands as C m1 m 2m 1 A m1 m 2m 1 B [m1 m 2m 1] Thus, we have AB A 12m B [12m] + A 23m1 B [23m1] + + which is the required result A m 1m1m 3m 2 B [m 1m1m 3m 2] + A m1 m 2m 1 B [m1 m 2m 1], j1

5 On Hadamard and Kronecker products over Matrix of Matrices 17 2I O Example 32 Let A I 3I 2I I and B I O M 2 M 2 K, then 4I 2I AB 5I I 2I O 2I O O 2I + I 3I 2I O 3I I A 12 B [12] + A 21 B [21] Example 33 Let A A ij and B B ij M 3 M l K, then AB I I I I A 11B 11 + A 12 B 21 + A 13 B 31 A 11 B 12 + A 12 B 22 + A 13 B 32 A 11 B 13 + A 12 B 23 + A 13 B 33 A 21 B 11 + A 22 B 21 + A 23 B 31 A 21 B 12 + A 22 B 22 + A 23 B 32 A 21 B 13 + A 22 B 23 + A 23 B 33 A 31 B 11 + A 32 B 21 + A 33 B 31 A 31 B 12 + A 32 B 22 + A 33 B 32 A 31 B 13 + A 32 B 23 + A 33 B 33 A 11 A 12 A 13 A 21 A 22 A 23 A 31 A 32 A 33 B 11 B 22 B 33 B 11 B 22 B 33 B 11 B 22 B 33 A 13 A 11 A 12 B 31 B 12 B 23 + A 23 A 21 A 22 B 31 B 12 B 23 A 33 A 31 A 32 B 31 B 12 B 23 A 123 B [123] + A 231 B [231] + A 312 B [312] + A 12 A 13 A 11 A 22 A 23 A 21 A 32 A 33 A 31 B 21 B 32 B 13 B 21 B 32 B 13 B 21 B 32 B 13 A comparable expansion of AB when the MMs A and B are not square can be given by the following corollary Corollary 34 Let A A ij M m k M l K and B B ij M k m M l K, with m k, then, AB s A s B [s] S AIs B[s], where the summation runs over all cyclic permutations s s 1 s 2 s m consisting of all the first m indices of S 12mk For a particular s s 1 s 2 s m S 12mk, A s is as given in 2 with k replaced by m and B [s] is as in 3 Example 35 Let us take A11 A A 12 A 13 A 21 A 22 A 23 and B B 11 B 12 B 21 B 22 B 31 B 33, Here, S 123 {123, 231, 312} as usual, but the permutations that we consider have only the first two parts, ie, s 12, 23, or 31 Then, AB 4 Kronecker Product A11 B 11 + A 12 B 21 + A 13 B 31 A 11 B 12 + A 12 B 22 + A 13 B 32 A 21 B 11 + A 22 B 21 + A 23 B 31 A 21 B 12 + A 22 B 22 + A 23 B 32 A 12 B [12] + A 23 B [23] + A 31 B [31] A11 A 12 B11 B 22 A12 A + 13 A 21 A 22 B 11 B 22 A 22 A 23 A13 A + 11 B31 B 12 A 23 A 21 B 31 B 12 B21 B 32 B 21 B 32 In this section, we give a definition of Kronecker product over MMs, and some properties

6 18 ZKishka et al Definition 41 Let A A ij M n m M l K and B B ij M p q M l K, the np mq MMs is called Kroncker product of A and B A B A 11 B A 12 B A 1m B A 21 B A 22 B A 2m B A n1 B A n2 B A nm B In order to explore the variety of applications of the Kronecker product we introduce the notation of the VMs operator, Definition 42 For any A M m n M l K, the VMs-operator is defined as VMs A A 11,, A m1,, A 1n,, A mn, ie, the entries of A are stacked column wise forming a vector of matrices of length mn Example 43 Let 2 0 A , and 2 1 B 1 2, then A B Properties of the Kronecker Product Theorem 44 Let A M m n M l K, B M r s M l K, C M n p M l K and D M s t M l K, then A B C D AC BD Proof Simply verify that A 11 B A 1n B C 11 D C 1p D A B C D A m1 B A mn B C n1 D C np D n k1 A n 1kC k1 BD k1 A 1k C kp BD n k1 A n mkc k1 BD k1 A mk C kp BD AC BD

7 On Hadamard and Kronecker products over Matrix of Matrices 19 Theorem 45 Let A M m n M l K and B M r s M l K, then A B t A t B t Proof Suppose that A A ij, B B kl and simply verify using the definition kronecker product over MMs A B t A 11 B A 1n B A m1 B A mn B A 11 B t A m1 B t A 1n B t A nm B t A t B t t Corollary 46 Let A M m n M l K and B M r s M l K be MMs symmetric, then A B is symmetric Proof Suppose that A A ij and B B kl are symmetric MMs and A ij, B kl M l K, then A B t A t B t A B So A B is symmetric Theorem 47 If A and B are nonsingular, then A B 1 A 1 B 1 Proof Using the above Theorem 9, simply note that A B A 1 B 1 I I I Theorem 48 Suppose that E M l K, A M m n M l K and B M r s M l K, then EA B A EB E A B, where M l K is commutative ring with unity Proof Since EA B EA 11 B EA 1n B EA m1 B EA mn B A 11 EB A 1n EB A m1 EB A mn EB A EB A 11 B A 1n B E A m1 B A mn B Remark 49 The above Theorem 13 is also true if we replace E by c K Remark 410 For all A and B we have; A B B A which can be justified by the following example

8 20 ZKishka et al and then Example 411 Let 2 0 A , B 1 2, B A B 2 0 B 0 2 A A B A Corollary 412 Let A M m M l K and B M n M l K, then 1 T r A B T r A T r B T r B A 2 det A B [det A] m [det B] n det B A 2 0 B 0 2 B 2 1 A 1 2 A Definition 413 Let A M m M l K and B M n M l K, then the Kronecker sumor tensor sum of A and B, denoted A B is the mn mn MMs I n A + B I m then Example 414 Let A Remark 415 In general, A B B A I 2I 3I 3I 2I I I I 4I 2I I and B 2I 3I, A B I 2 A + B I 3 3I 2I 3I I O O 3I 4I I O I O I I 6I O O I 2I O O 4I 2I 3I O 2I O 3I 5I I O O 2I I I 7I Lemma 416 The Kronecker product over MMs is associative, ie, A B C A B C, where A M m n M l K, B M r s M l K and C M s t M l K Lemma 417 The Kronecker product over MMs is right distributive, ie, where A, B M m n M l K and C M s t M l K A + B C A C + B C, Lemma 418 The Kronecker product over MMs is left distributive, ie, where A M m n M l K, B and C M s t M l K A B + C A B + A C,

9 On Hadamard and Kronecker products over Matrix of Matrices Application The Kronecker product can be used to present linear matrix equations in which the unknowns are MMs Examples for such equations are: AX B, AX + X B C, AX B C These equations are equivalent to the following systems of equations: I A VMs X VMs B, [ I A + B t I ] VMs X VMs C, B t A VMs X VMs C 43 Relation between Hadamard and Kronecker product It is clear that the Hadamard product of two MMs is the principal sub-mms of the Kronecker product of the two MMs This relation can be expressed in an equation as follows Lemma 419 For A and B M m n M l K, then we have A B Z t 1 A B Z 2, where Z 1 is the selection MMs of order m 2 m and Z 2 is the selection MMs of order n 2 n Example 420 Let I 2I A O I 2I I, B I O where A and B M 2 M 2 K, and the selection MMs I O O O Z 1 Z 2 O O O I,, then A B Z t 1 A B Z 2 5 Conclusion In this work, we have introduced Hadamard and Kronecker product over a new algebraic structure which is called MMs and some of their properties It is worth to mention that the new concepts are applicable in many different branches of mathematics such as group theory, statistics, combinatorics including graphs, other discrete structures, and functional analysis Further research on this topic is now under investigation and will be reported in forthcoming works References [1] R Horn, Topics in Matrix Analysis, Cambridge University Press, 1994 [2] C Johnson, Matrix Theory and Applications, American Mathematical Society, 1990 [3] G P H Styan, Hadamard products and multivariate statistical analysis, Linear Algebra Appl, 6, 1973, [4] H Neudecker and S Liu, A Kronecker matrix inequality with a statistical application, Econometric Theory, , 655

10 22 ZKishka et al [5] H Neudecker and A Satorra, A Kronecker matrix inequality with a statistical application, Econometric Theory, , 654 [6] G Trenkler, A Kronecker matrix inequality with a statistical application, Econometric Theory, , [7] C R Rao and M B Rao, Matrix Algebra and Its Applications to Statistics and Econometrics, World Scientific, Singapore, 1998 [8] F Zhang, Matrix Theory Basic Results and Techniques, Springer, New York, 1999 [9] S Liu, On matrix trace Kantorovich-type inequalities, In: Innovations in Multivariate Statistical Analysis- A Festschrift for Heinz Neudecker Ed RDH Heijmans, DSG Pollock and A Satorra, Kluwer Academic Publishers, Dordrecht, 2000a, pp [10] S Liu, Inequalities involving Hadamard products of positive semidefinite matrices, J Math Ana Appl, 243, 2000b, [11] H L V Trees, Detection, Estimation, and Modulation Theory, Part IV, Optimum Array Processing, Wiley, 2002 [12] MW Browne, Generalized least squares estimators in the analysis of covariance structures, South African Statist J, , 1-24 [13] M Faliva, Identificazione Stima nel Modello Lineare ad Equazioni Simultanee, Vita e Pensiero, Milan, Italy, 1983 [14] F Pukelsheim, On Hsu s model in regression analysis, Statistics, 8, 1977, [15] CG Khatri and C R Rao, Solutions to some functional equations and their applications to characterization of probability distributions, Sankhya, 3968, [16] C R Rao, Estimation of heteroscedastic variances in linear models, J Amer Statist Assoc, , [17] C R Rao and M B Rao, Matrix Algebra and Its Applications to Statistics and Econometrics, World Scientitifc, Singapore, 1998 [18] R A Horn and R Mathias, Block-matrix generalizations of Schur s basic theorems on Hadamard products, Linear Algebra Appl, , [19] R P Singh, Some Generalizations in Matrix Differentiation with Applications in Multivariate Analysis, PhD Thesis, University of Windsor, 1972 [20] D S Tracy and R P Singh, A new matrix product and its applications in matrix differentiation, Statist Neerlandica, , [21] D S Tracy and K G Jinadasa, Partitioned Kronecker products of matrices and applications, Canada J Statist, , [22] Z Kishka, M Saleem, M Abul-Dahab and A Elrawy, On the algebraic structure of certain matrices, submitted to J of the Egyptian Math Soc, in revised version, 2017