TRACES OF HADAMARD AND KRONECKER PRODUCTS OF MATRICES. 1. Introduction

Transcription

1 Math Appl , DOI: /ma TRACES OF HADAMARD AND KRONECKER PRODUCTS OF MATRICES PANKAJ KUMAR DAS ad LALIT K VASHISHT Abstract We preset some iequality/equality for traces of Hadamard product ad Kroecker product of matrices Some umerical examples are give to support the results 1 Itroductio The Hadamard or Schur ad Kroecker products are widely studied ad applied i matrix theory, statistics, system theory ad other areas [2] It was Schur who iitially studied algebraic ad aalytic properties of Hadamard product I 1990, Hor [1] presets a widespread iformatio focusig o the Hadamard product Magus ad Neudecker [4] give some basic results ad statistical applicatios ivolvig Hadamard or Kroecker products For basics o these two matrix products, oe may refer to [3,5 7] I [6], the authors ivestigated traces of Hadamard ad Kroecker products of matrices ad obtaied some iequalities for traces of products of matrices I this ote, we preset some iequality for traces of Hadamard product, Kroecker product ad mixed type product of matrices We recall the basic defiitios ad otatios to make our presetatio selfcotaied The set of all positive real umbers is deoted by R + By M we deote the family of -by- matrices over the real field R For A = [a ij ] M, the scalar a ii is called the trace of A ad is usually deoted by tra or tracea The Hadamard product of two matrices A = [a ij ] ad B = [b ij ] of idetical size is just their elemet-wise product which is give by A B = [a ij b ij ] m Let A = [a ij ] be a m-by- matrix ad let B = [b ij ] be a p-by-q matrix The Kroecker product of A ad B is defied as A B = a 11 B a 12 B a 1 B a 21 B a 22 B a 2 B a m1 B a m2 B a m B MSC 2010: primary 15A45; secodary 15A24 Keywords: trace, Hadamard product, Kroecker product 143

2 144 P K DAS ad L K VASHISHT Lemma 11 [6] Let A M ad B M m The, tra B = trb A = tratrb Theorem 12 [6] If A = [a ij ], B = [b ij ] M, the 1 tra B = tra B 1 j=i+1 a ii a jj b ii b jj 2 tra B tr A+B 2 A+B 2 2 Mai results We start with a iequality ivolvig the trace of m times Hadamard product of a matrix ad the trace of the give matrix Propositio 21 Let A M with positive real diagoal The, tr A A A A m tra m Proof Let A = [a ij ] M ad let a ii R + The, by the defiitios of Hadamard product ad trace of matrices, we have m m tr A A A A = a m ii a ii = tra m m The propositio is proved Propositio 22 If A = [a ij ], B = [b ij ] M, the tr A A A B B B 1 = b ii a ii j=i+1 Proof Sice tr A A A B B B = applyig Theorem 1 i [6], we get tr A A A B B B 1 = b ii a ii j=i+1 a ii a jjb ii b jj a ii b ii, a ii a jjb ii b jj The followig propositio provides a relatio betwee the trace of a matrix ad Kroecker product of the give matrix

3 TRACES OF HADAMARD AND KRONECKER PRODUCTS OF MATRICES 145 Propositio 23 For ay A M, we have tr A A A A m = tra m Proof We prove the propositio by iductio By Lemma 11, the equatio is true for = 2 Assume that it is true = k, ie, tr A A A A k = tra k 21 Agai usig Lemma 11, we compute tr A A A A k+1 = tr The propositio is proved Example 24 Let A = The, A A = [ [ ] ], A A A = A A A A k = tra k tra by 21 = tra k+1 tra ad tra = 4 Now we have tra A = = tra 3 So, Propositio 21 is true Also, tra A A = 64 = tra 3 This verifies Propositio 22 Propositio 25 For ay A M, we have tr A + A t A + A t = 4 tra A Proof Let A = [a ij ] M The tr A + A t A + A t = a ii + a ii a ii + a ii = 4 a 2 ii = 4 tra A The propositio is proved Propositio 26 For ay A, B M, we have tr A + B A B = tra A trb B

4 146 P K DAS ad L K VASHISHT Proof Let A = [a ij ], B = [b ij ] M be arbitrary We compute tr A + B A B = a ii + b ii a ii b ii = a ii 2 b ii 2 = tra A trb B Propositio 27 For ay A, B M, we have tr A + B A + B = tra A + 2trA B + trb B Proof Similar to proof of Propositio 26 The followig propositio gives the relatioship betwee trace of a matrix obtaied as Kroecker product of a fiite sum of matrices ad the traces of the matrices We prove the result for two matrices Propositio 28 Let A, B M The tr A + B A + B A + B A + B k k k i i k = tra trb i Proof Usig Propositio 23, we compute tr A + B A + B A + B A + B k = tra + B = tra + trb = k The result is proved k i i k tra trb i Remark 29 Usig Propositio 21, for ay A, B M with positive diagoal, we ca show that tr A + B A + B A + B A + B k k i i tra trb i

5 TRACES OF HADAMARD AND KRONECKER PRODUCTS OF MATRICES 147 The ext propositio gives a trace iequality for the Hadamard product of matrix sums Propositio 210 If A, B M, the tra B 2 1 A 16 tr + B A + B A + B A + B Proof Usig Theorem 12 ad Propositio 23, we compute A + B A + B A + B A + B tra B 2 tr tr A + B A + B A + B A + B = tr = 1 16 tr A + B A + B A + B A + B The propositio is proved The followig theorem provides a relatio betwee the trace of matrices geerated by Kroecker ad Hadamard product of matrices Theorem 211 If A, B, C, D M, the 1 A B C D = A C B D 2 tra B C = tra B C 1 j=i+1 a ii a jj b ii c ii b jj c jj Proof Let us write A = [a ij ] ad C = [c ij ] The, we compute A B C D a 11 B a 12 B a 1 B c 11 D c 12 D c 1 D a 21 B a 22 B a 2 B = c 21 D c 22 D c 2 D a 1 B a 2 B a B c 1 D c 2 D c D a 11 c 11 B D a 12 c 12 B D a 1 c 1 B D a 21 c 21 B D a 22 c 22 B D a 2 c 2 B D = = A C B D a 1 c 1 B D a 2 c 2 B D a c B D Hece, 1 is proved Let us write B = [b ij ] The, by Theorem 12, we have 1 tra B C = tra B C Thus, 2 is proved 1 = tra B C j=i+1 j=i+1 a ii a jj b ii c ii b jj c jj a ii a jj b ii c ii b jj c jj

6 148 P K DAS ad L K VASHISHT Corollary 212 If A, B, C, D M, the tr A B C D = tra B C D 1 Ideed, by Theorem 211, we ca write tr A B C D j=i+1 Thus, by Theorem 12, we have tr A B C D = tra C B D 1 a ii c ii a jj c jj b ii d ii b jj d jj = tr A C B D j=i+1 a ii c ii a jj c jj b ii d ii b jj d jj Remark 213 If A, C M ad B, D M k k, the the equatios give i Theorem 211 hold too [ ] [ ] [ ] Example 214 Let A =, B =, C = ad [ ] D = 1 3 The A B = , C D = ; [ ] [ ] A C =, B D = ; ad [ ] A B D = , A B D = It is easy to see that A B C D = = A C B D We compute 1 tra B D j=i+1 a ii a jj b ii d ii b jj d jj

7 TRACES OF HADAMARD AND KRONECKER PRODUCTS OF MATRICES 149 = here = 2 = 28 = tr A B D Thus, Theorem 211 is verified To coclude the paper, we give a result that coects the traces of matrices obtaied by Kroecker ad Hadamard product of matrices i terms of the trace of Hadamard product of matrices Theorem 215 If A i, B i M for 1 i, the tr A 1 A 2 A B 1 B 2 B = tra i B i Proof We prove the theorem by iductio Clearly, the equatio is true for = 1 Assume that it is true for = k, ie, k tr A 1 A 2 A k B 1 B 2 B k = tra i B i We compute tr A 1 A 2 A k+1 B 1 B 2 B k+1 { = tr A 1 A 2 A k B 1 B 2 B k A k+1 B k+1 by Theorem 211 = tr A 1 A 2 A k B 1 B 2 B k tra k+1 B k+1 by Lemma 11 k = tra i B i tra k+1 B k+1 tra i B i k+1 = Thus, the equatio is true for = k + 1 The theorem is proved Remark 216 Theorem 215 is useful because the computatio of A i B i 1 i is much easier tha A 1 A 2 A B 1 B 2 B Refereces [1] R A Hor, The Hadamard product, Proc Symp Appl Math , [2] R A Hor ad C R Johso, Matrix Aalysis, 2d ed, Cambridge Uiversity Press, 2010 [3] S Liu ad G Trekler, Hadamard, Khatri Rao, Kroecker ad other matrix products, It J If Syst Sci , [4] J R Magus ad H Neudecker, Matrix Differetial Calculus with Applicatios i Statistics ad Ecoometrics, 2d ed, Wiley, Chichester, UK, 1999 [5] M Özel, Trace ad determiat iequalities for Hadamard products, Iteratioal Joural of Physical Scieces , 1 4 [6] N Taskara ad I H Gumus, O the traces of Hadamard ad Kroecker products of matrices, Selcuk Joural of Applied Mathematics , 31 36

8 150 P K DAS ad L K VASHISHT [7] H Zhag ad F Dig, O the Kroecker products ad their applicatios, J Appl Math , Article ID , 8 pp Pakaj Kumar Das, Departmet of Mathematical Scieces, Tezpur Uiversity, Napaam, Tezpur, Assam , Idia O lie from Shivaji College, Uiversity of Delhi, Departmet of Mathematics, Raja Garde, Rig Road, New Delhi , Idia pakaj4thapril@yahoocoi, pakaj4@tezuereti Lalit K Vashisht, Departmet of Mathematics, Uiversity of Delhi, Delhi , Idia lalitkvashisht@gmailcom