Trace Inequalities for a Block Hadamard Product

Transcription

1 Filomat 32:1 2018), Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: Trace Inequalities for a Block Hadamard Product Saliha Pehlivan a, Mustafa Özel b, Cenap Özel c a Department of Biomedical Engineering, Faculty of Engineering and Architecture Gediz University Izmir, Turkey b Department of Geophysics Engineering, Faculty of Engineering Dokuz Eylul University Izmir, Turkey c Department of mathematics, Faculty of Sciences King Abdulaziz University Jeddah, Saudi Arabia And Department of mathematics, Faculty of Science Dokuz Eylul University Izmir, Turkey Abstract In this paper, some inequalities for the trace and eigenvalues of a block Hadamard product of positive semidefinite matrices are investigated In particular, a Hölder type inequality and inequalities related to norm and determinants of block matrices are obtained Additionally, the relation between the trace of block Hadamard product and the usual Kronecker product is established 1 Introduction Block Kronecker and block Hadamard products seem to be firstly defined and used by Horn, Mathias and Nakamura in 1991[4]) Günther and Klotz presented a survey focusing on the study of a block Hadamard and block Kronecker products of positive semidefinite matrices in 2012[2]) In that paper, some properties of such products were discussed and an inequality a lower bound) on the determinant of the block Hadamard product was given In 2014, Lin provided an Oppenheim type inequality for the determinant of the block Hadamard product[5]), which confirms a conjecture in [2] So far, there have not been quite many results on the trace of such products in the literature In this paper, we shall be mainly interested in the inequalities for the trace of the block Hadamard product For the rest of the paper, we first introduce the definitions and terminology that are used throughout the paper In Section 3, we give some lemmas that play important roles in our results In Section 4, we give our main results on the upper and lower bounds for the trace of the block Hadamard product of positive definite matrices 2010 Mathematics Subject Classification Primary 42C15; Secondary 46C05, 47B10 Keywords Block Hadamard Product, Trace, Positive Semi)Definite Matrices, Majorization, Norm Received: 06 March 2017; Accepted: 22 April 2017 Communicated by Fuad Kittaneh addresses: salihapehlivan@gedizedutr Saliha Pehlivan), mustafaozel@deuedutr Mustafa Özel), cenapozel@gmailcom Cenap Özel)

2 S Pehlivan, M Özel, C Özel / Filomat 32:1 2018), Preliminaries Let M m n be the linear space of m n matrices with complex entries and M p,q M m,n ) be the space of p q block matrices, and write M n,n : M n and M p : M p,p M n,n ) The identity matrix in M p is denoted by I p dia I n,, I n ) where I n M n A matrix A M p is Hermitian if A A where A is the conjugate transpose of A A Hermitian matrix A is said to be positive definite positive semidefinite), denoted by A > 0 A 0), if x Ax > 0 x Ax 0) for all nonzero x C The eigenvalues and singular values of A M p are denoted by λ 1 A), λ 2 A),, λ pn A) and σ 1 A), σ 2 A),, σ pn A), respectively They are arranged in decreasing order: for a Hermitian matrix A M p, λ 1 A) λ 2 A) λ pn A), and σ 1 A) σ 2 A) σ pn A) Note that σ i A) λ i A) for a positive semidefinite matrix A for i 1,, pn since σ i A) λ i A A) 1/2 Let A A ij ) M p,q and B B ij ) M s,t Then we call matrices A, B block commuting if every n n block of A commutes with every n n block of B Let A A ij ) M p,q M m,l ) and B B ij ) M p,q M l,n ) Then the block Hadamard product of A and B is defined by A B A ij B ij ) If A and B are both positive definitepositive semidefinite) and block commuting matrices, then A B is positive definite positive semidefinite) as in [2] Corollary 33 Let A M m,l and B B ij ) M s,t M l,n ) Then the block Kronecker product of A and B is defined by A B AB ij ),,t,,s where AB ij is the usual matrix product of A and B ij For A A ij ) M p,q M m,l ), the block Kronecker product is given by A B A ij B),,q,,p For real vectors x x 1,, x n ) and y y 1,, y n ) in decreasing order, it is said that x is majorized by y x y) if n n n n x i y i for k 2,, n and x i y i ik ik A linear map Φ from M n to M n is called doubly stochastic if it is positive A 0 ΦA) 0), unital ΦI) I) and trace preserving tr ΦA)) tra) Then, for A M pn and Φ : M pn M pn, it is clear that ΦA) A I is doubly stochastic Thus, by Lemma 214 in [8], we have λa I) λa) 1) 3 Lemmas In this section, we give lemmas that are fundamental in our main results Lemma 31 [3] For any two matrices A and B, σab) lo σa)σb) Lemma 32 [6] Let α i > 0 i 1,, n) and n α i 1 Let a ij > 0 for j 1,, m Then m a α 1 1j aα 2 2j aα n m ) α1 m ) α2 a nj 1j a 2j m ) αn a nj Lemma 33 [7] Let A i, B i M n C) for i 1,, m be positive semidefinite matrices and p, q be positive real numbers such that 1/p + 1/q 1 Then tr m ) m A i B i tr A p i )) 1/p m )) 1/q tr B q i For any A, B M p, A B A B)[α] where α is a block index set such that α {1, p + 2, 2p + 3,, p 2 }, there exists a unital positive linear map Φ from M p 2 to M p such that ΦA B) A B for all A, B M p as in [8] Lemma 19

3 Lemma 34 For any A, B > 0, we have S Pehlivan, M Özel, C Özel / Filomat 32:1 2018), log A + log B) I loga B) A, B > 0 2) Proof First note that ΦA I) A I I A ΦI A) and A I) k A k I) by Proposition 23 of [2] so for any analytic function f we have f A I) f A) I by power series representation of f Then, since Φ is linear, we get log A + log B) I Φ log A + log B) I ) 3) Φ log A I + log B I ) 4) Φ log A I ) + Φ log B I ) 5) Φ log A I ) + Φ I log B ) 6) Φ log A I + I log B ) 7) Φ loga I) + logi B) ) 8) Φ loga I)I B)) ) 9) Φ loga B ) 10) log ΦA B ) 11) loga B) 12) where 10) and 11) follow from the facts, respectively, that A I)I B) A B see [2]) and Φlog A) logφa)) for a unital positive linear map Φ and A > 0 see [8]) 4 Main Results In this section, we first introduce an inequality on the product of eigenvalues of the block Hadamard product of positive definite matrices that generalize Oppenheim s inequality Then we give some upper and lower bounds on the trace of block Hadamard product of positive definite positive semidefinite) matrices Theorem 41 Let A, B M p be block commuting and positive definite matrices Then pn λ j A B) pn λ j AB), k 1, 2,, pn 13)

4 S Pehlivan, M Özel, C Özel / Filomat 32:1 2018), Proof By 2) and 1), respectively, for all k 1, 2,, pn we have log pn λ j A B) ) pn pn pn pn λ j loga B) ) λ j log A + log B) I ) λ j log A + log B ) λ j log A 1/2 BA 1/2) log pn λ j AB) ) where the last inequality follows from log A + log B loga 1/2 BA 1/2 ) see [8] pg 22) We remark here that, by Theorem 41 for k1, we get Openheim s inequality for the block Hadamard product: deta B) det A det B The following result is a Hölder type inequality for the trace of block Hadamard product Theorem 42 Let A A ij ), B B ij ) M p be positive semidefinite block matrices and q 1, q 2 1/q 1 + 1/q 2 1 Then > 1 with i) tra B) tra q 1 ) tra B) tra q 1 ) 1/q1 ) 1/q2 trb q 2 ) 1/q1 ) q 1/q2 trb 2 Proof i) By Lemma 33 and the linearity of the trace we have tra B) ) tra B ) tr A B tr A q 1 1/q1 tr 1/q 1 tra q 1 trb q 2 ) For m 1 in Lemma 33, we obtain tra B) tra B ) tra q 1 ) 1/q1 ) q 1/q2 trb 2 B q 2 1/q2 1/q 2

5 S Pehlivan, M Özel, C Özel / Filomat 32:1 2018), The relation between the trace of block Hadamard product and the trace of Kronecker product is given in the following form: Corollary 43 Let A, B M 2 be block commuting and positive semidefinite matrices Then tra B) tr A A) B B)) ) 1/2 Proof Let p q 1 q 2 2 in Theorem 42i) Then we get tra B) 2 Moreover, we have tr 2 ) 1/2 2 tra 2 trb 2 )) 1/2 tr 2 A 2 B 2 ) 1/2 )) 1/2 tr A A2 22 )) 1/2 tr B B2 22 tra A) ) 1/2 ) 1/2 trb B) tra A)trB B) ) 1/2 tr A A) B B)]) ) 1/2, where the last equality follows from the fact that trn M) trntrm Thus, we obtain the result )) 1/2 This can also be generalized to the following: tra B) tr p A) p B)) ) 1/2 for A, B M p and A, B > 0, where p A denotes for the block Hadamard product of A p times Now, an inequality that depends on the norm and trace of block matrices is given as below Theorem 44 Let A A ij ), B B ij ) M p be block commuting and positive semidefinite matrices Then for any natural number k 1, we have tr { k A B)) min A k trb k, } B k tra k, where is a matrix norm In particular, tra B) min { p A trb, } B tra Proof First note that we have A k Bk A B ) k since A and B are block commuting By Lemma 31 and

6 S Pehlivan, M Özel, C Özel / Filomat 32:1 2018), λ j A) A, we get tr k A B)) tr p A B ) k) n λ k j A B ) n A k λ k j B ) A k n λ k j B ) A k trb k Similarly, we have tr k A B)) B k tra k Corollary 45 Let A A ij ), B B ij ) M p be block commuting and positive semidefinite matrices Then we have the followings: i) For q 1 + q 2 1 and q 1, q 2 > 0, tr k A B)) { min A ) q 1/q 1 p ) q2 ) 1/q2 1k trb k, p ) tr [ k A] min { n A k, n tra k } B q 1k ) 1/q 1 p ) q2 ) 1/q2 } tra k Proof i) follows from Theorem 44 and Theorem 28 of [7] Letting B I in Theorem 44, we get ) Finally, we shall discuss a lower bound on the trace of block Hadamard product in terms of determinants of block matrices Theorem 46 Let A, B M p be positive definite block matrices Then tr k A B)) n deta B ) ) k/n In particular, tra B) n p deta B ) ) 1/n Proof For each A we can find a unitary matrix U i such that U i Λ U i A since A > 0 where Λ dia λ 1 Λ ),, λ n Λ )) Define b i) jj as the diagonal elements of U i Bk U i for j 1,, n Then by arithmetic-

7 geometric mean inequality we get tr k A B)) tra B ) k ) n n n S Pehlivan, M Özel, C Özel / Filomat 32:1 2018), tru i Λ k U i Bk ) trλ k U i B U i ) k ) λ k 1 Λ )b i) λk nλ )b i) nn λ k 1 Λ )b i) 11 ) λk nλ )b i) 1/n nn detλ k )) 1/n detu i B U i ) k) 1/n ) k/n deta detb where the last inequality follows from the fact that detc c 11 c 22 c nn for any matrix C M n with diagonal elements c, i 1,, n 5 Example Let A B be positive definite matrix such that ) A11 A 12 A A Then , , 49058, are the eigenvalues of A A For the result of Theorem 41, we have the following table: k1 k2 k3 k4 4 λ i A A) ik 4 λ i AA) ik In Theorem 42, for q 1 q 2 2, we have the equality in both cases On the other hand, for q 1 3, q 2 3/2 we have and , respectively, as an upper bound for tra A) 69 For k 1 and k 2 in Theorem 44, we have the following inequalities 69 < 83 and 1953 < 2165, respectively

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