n Inequalities Involving the Hadamard roduct of Matrices 57 Let A be a positive definite n n Hermitian matrix. There exists a matrix U such that A U Λ

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1 The Electronic Journal of Linear Algebra. A publication of the International Linear Algebra Society. Volume 6, pp , March ISSN http//math.technion.ac.il/iic/ela ELA N INEQUALITIES INVLVING THE HADAMARD RDUCT F MATRICES Λ B. MND y AND J. E CARIĆz Abstract. Recently, the authors established a number of inequalities involving integer powers of the Hadamard product of two positive definite Hermitian matrices. Here these results are extended in two ways. First, the restriction to integer powers is relaxed to include all real numbers not in the open interval (1; 1). Second, the results are extended to the Hadamard product of any finite number of Hermitian positive definite matrices. Key words. matrix inequalities, Hadamard product AMS subject classifications. 15A45 1. Introduction. Let A B be n n matrices. A ffi B denotes the Hadamard product A Ω B the Kronecker product of A B. These two products are related by the following relation [2], [3]. There exists an n 2 n selection matrix J such that J T J I A ffi B J T (A Ω B)J Note that J T is the n n 2 matrix [E 11 E 22 E nn ], where E ii is the n n matrix of zeros except for a one in the (i; i)th position. Using this result, in [4] the authors proved a number of inequalities involving integer powers of the Hadamard product of two positive definite Hermitian matrices. Here we extend these results in two ways. First, the restriction to integer powers is relaxed to include all real numbers not in the open interval (1; 1). Second, the results are extended to the Hadamard product of any finite number of n n Hermitian positive definite matrices. 2. Notation reliminary Results. The Hadamard Kronecker products of matrices ; i 1;k, will be denoted by ffi k, respectively. k We shall make frequent use of the following property of the Kronecker product (AB) Ω (CD)(A Ω C)(B Ω D) For a finite number of matrices ; B i ; i 1;;k, this becomes (1) Ω B i ( Ω B i ) Λ Received by the editors on 11 ctober Accepted for publication on 25 January Hling editor Daniel Hershkowitz. y Department of Mathematics, La Trobe University, Bundoora, Victoria 3083, Australia (matbm@lurac.latrobe.edu.au). z Faculty of Textile Technology, University of Zagreb, Zagreb, Croatia, Applied Mathematics Department, University of Adelaide, Adelaide, South Australia 5005, Australia (jpecaric@maths.adelaide.edu.au). 56

2 n Inequalities Involving the Hadamard roduct of Matrices 57 Let A be a positive definite n n Hermitian matrix. There exists a matrix U such that A U Λ [ 1 ; 2 ;; n ]U; U Λ U I; where [ 1 ; 2 ;; n ] is the diagonal matrix with i, the positive eigenvalues of A, along the diagonal [1]. For any real number s; A s is defined by A s U Λ [ s 1 ;s 2 ;;s n]u Lemma 2.1. Let A B be positive definite Hermitian n n matrices s a nonzero real number. Then (2) A s Ω B s (A Ω B) s roof. Assume where fl i are the eigenvalues of B. Then Note that B V Λ [fl 1 ;fl 2 ;;fl n ]V; V Λ V I; A s Ω B s (U Λ [ s 1;; s n]u) Ω (V Λ [fl s 1;;fl s n]v ) (U Λ Ω V Λ )([ s 1 ;;s n] Ω [fl s 1 ;;fls n])(u Ω V ) (U Ω V ) Λ ([ s 1 ;;s n] Ω [fl s 1 ;;fls n])(u Ω V )(A Ω B) s (U Ω V ) Λ (U Ω V ) (U Λ Ω V Λ )(U Ω V ) (U Λ U) Ω (V Λ V )I Ω I I n 2 Equation (2) extends readily, for a finite number of n n positive definite Hermitian matrices ; i 1;;k,to (3) k (A s i ) s Lemma 2.2. Let ; i 1;;k, be n n matrices. There exists an n k n selection matrix such that T I (4) We prove this for three matrices. 1 A ffi B ffi C ψ k k ffi T The extension from m to m + 1 is similar. A ffi J T (B Ω C)J J T (A Ω (J T (B Ω C)J))J J T ((IAI) Ω (J T (B Ω C)J))J J T (I Ω J T )(A Ω B Ω C)(I Ω J)J by (1) J T (I Ω J) T (A Ω B Ω C)(I Ω J)J ^J T (A Ω B Ω C) ^J; 1 This proof was provided by George Visick in a private communication.

3 58 B. Mond J. ecarić where ^J is the n 3 n matrix ^J (I Ω J)J. Note that ^J T ^J I. 3. Results. In this section, ; i 1;;k, will denote n n positive definite Hermitian matrices. A j means that A j is positive semidefinite. Theorem 3.1. Let r s be real numbers r<s, either r 2 (1; 1) s2 (1; 1) or s 1 r 1 2 or r 1 s 1 2. Then (5) As i 1s Ar i 1r roof. We make use of the following result [5]. Let A be an n n positive definite Hermitian matrix let V be an n t matrix such that V Λ V I. Then (V Λ A s V ) 1s (V Λ A r V ) 1r for all real r s, r<s, such that either r 2 (1; 1) s2 (1; 1) or s 1 r 1 2 or r 1 s 1. 2 Here instead of V,we use the n k n selection matrix given by (4). Noting (3), we have 1s N As i t k As i N s t k 1s A i N t k 1r Ar i kffi Some special cases of (5) are the following 1 or, equivalently For r>1, we have or, equivalently, A1 i 1s T N k Ar i 1r 1 A kffi i A1r i For r 2, the last two inequalities become A1 i Ar i 1r 1r A2 i 12 r 1r

4 n Inequalities Involving the Hadamard roduct of Matrices 59 A12 i 12 Theorem 3.2. Let r s be nonzero real numbers such that s > r s2 (1; 1) or r 2 (1; 1). Then (a) As i 1s μ 4 Ar i 1r ; where (6) fl Mm, k. Also, ρ r(fl ff μ4 s fl r 1s ρ ) s(fl ff r fl s 1r ) (s r)(fl r 1) (r s)(fl s ; 1) M m are, respectively, the largest smallest eigenvalues of (b) As i 1s Ar i 1r 4I; where (7) 4 max f[ M s +(1 )m s ] 1s [ M r +(1 )m r ] 1r g 2[0;1] roof. Let A be an n n positive definite Hermitian matrix with eigenvalues contained in the interval [m; M], where 0 <m<m, let V be an n t matrix such that V Λ V I. If r s are nonzero real numbers such that r<s either s2 (1; 1) or r 2 (1; 1), then [6] (8) (V Λ A s V ) 1s μ 4(V Λ A r V ) 1r where μ 4 is given by (6), (9) (V Λ A s V ) 1s (V Λ A r V ) 1r 4I where 4 is given by (7). Thus for part (a), from (8) noting (3) (4), we have μ 4 " As i T 1s " T r #1r μ 4 " A s i 1s " ψ k # T T A r i s # 1r 4 μ k ffi Ar i #1s 1r

5 60 B. Mond J. ecarić For part (b), from (9), As i 1s Ar i " 1r " T T s #1s 1s " ψ k # T " r #1r T A s i A r i 4I # 1r Remark 3.3. The cases k 2 of the above results were also considered in [7] Special Cases. For s 2 r 1,we get 12 (M + m) A2 i 2 p kffi Mm A2 i 12 (M m)2 4(M + m) I For s 1 r 1, we get A (m + M)2 kffi 1 i 4Mm A1 i We note that the eigenvalues of 1 A 1 i ( p M p m) 2 I k are the n k products of the eigenvalues of ; i 1;;k. Thus, if the eigenvalues of ; i 1;;k, are ordered as i 1 i 2 i n > 0; i 1;;k; then the maximum minimum eigenvalues of m k i n. This leads to the following four inequalities A2 i i 1 + vut k i 1 i n i n 1 C A are M ; i 1

6 n Inequalities Involving the Hadamard roduct of Matrices 61 (10) A2 i 12 i A 1 i ψ k i 1 ψ k 4 i 1 i n i n 2 i vut k i n A1 i i n 2 1 ; I; i 1 v ut k 12 i n A I Finally, by taking A 1 i A1 i for in (10), we obtain i i 1 i n i n 2 1 The inequalities here are generalizations of those given in [4]. Additional inequalities of a similar kind are possible will be considered elsewhere. REFERENCES [1] F.E. Hohn. Elementary Matrix Algebra. MacMillan, New ork, [2] T. Kollo H. Neudecker. Asymptotics of eigenvalues unit-length eigenvectors of sample variance correlation matrices. J. Multivariate Anal., , [3] S. Liu H. Neudecker. Several matrix Kantorovich-type inequalities. J. Math. Anal. Appl., , [4] B. Mond J.E. ecarić. Inequalities for the Hadamard product of matrices. SIAM J. Matrix. Anal. Appl., , [5] B. Mond J.E. ecarić. n Jensen's inequality for operator convex functions. Houston J. Math., , [6] B. Mond J.E. ecarić. A matrix version of the Ky Fan inequalities of the Kantorovich inequality II. Linear Multilinear Algebra, , [7]. Seo, S.E. Takahasi, J. ecarić, J. Mićić. Inequalities of Furuta Mond-ecarić on the Hadamard product. J. Inequalities Appl., to appear.