Báo cáo hóa học: "EXTENSION AND GENERALIZATION INEQUALITIES INVOLVING THE KHATRI-RAO PRODUCT OF SEVERAL POSITIVE MATRICE" pot

EXTENSION AND GENERALIZATION INEQUALITIES INVOLVING THE KHATRI-RAO PRODUCT OF SEVERAL POSITIVE MATRICES ZEYAD ABDEL AZIZ AL ZHOUR AND ADEM KILICMAN Received 15 February 2005; Accepted 16 Oc tober 2005 Recently, there have been many authors, who established a number of inequalities in- volving Khatri-Rao and Hadamard products of two positive matrices. In this paper, the results are established in the following three ways. First, we find generalization of the inequalities involving Khatri-Rao product using results given by Liu (1999), Mond and Pe ˇ cari ´ c (1997), Cao et al. (2002), Chollet (1997), and Visick (2000). Second, we recover and develop some results of Visick. Third, the results are extended to the case of Khatri- Rao product of any finite number of matrices. These results lead to inequalities involving Hadamard product, as a special case. Copyright © 2006 Z. A. Al Zhour and A. Kilicman. This is an open access article distrib- uted under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Consider matrices A and B of order m ×n and p ×q, respectively. Let A = [A ij ]bepar- titioned with A ij of order m i ×n j as the (i, j)th block submatrix and let B = [B kl ]be partitioned with B kl of order p k ×q l as the (k,l)th block submatrix (m =  t i =1 m i , n =  d j =1 n j , p =  u k =1 p k , q =  v l =1 q l ). For simplicity, we say that A and B are compatible partitioned if A = [A ij ] t i, j =1 and B = [B ij ] t i, j =1 are square matrices of order m ×m and partitioned, respectively, with A ij and B ij of order m i ×m j (m =  t i =1 m i =  t j =1 m j ). Let A ⊗B, A ◦B, AΘB,andA ∗B be the Kronecker, Hadamard, Tracy-Singh, and Khatri-Rao products, respectively, of A and B. The definitions of the mentioned four matrix products are given by Liu in [5, 6]asfollows: (i) Kronecker product A ⊗B =  a ij B  ij , (1.1) where A = [a ij ], B = [b kl ] are scalar matrices of order m ×n and p ×q,respec- tively, a ij B is of order p ×q,andA ⊗B of order mp×nq; Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 80878, Pages 1–21 DOI 10.1155/JIA/2006/80878 2 Generalization inequalities for Khatri-Rao product (ii) Hadamard product A ◦B =  a ij b ij  ij = B ◦A, (1.2) where A = [a ij ], B =[b ij ] are scalar matrices of order m ×n, a ij b ij is a scalar, and A ◦B is of order m ×n; (iii) Tracy-Singh product AΘB =  A ij ΘB  ij =  A ij ⊗B kl  kl  ij , (1.3) where A = [A ij ], B = [B kl ] are partitioned matrices of order m ×n and p × q, respectively, A ij is of order m i ×n j , B kl of order p k ×q l , A ij ⊗B kl of order m i p k ×n j q l , A ij ΘB of order m i p ×n j q (m =  t i =1 m i , n =  d j =1 n j , p =  u k =1 p k , q =  v l =1 q l ), and AΘB of order mp×nq; (iv) Khatri-Rao product A ∗B =  A ij ⊗B ij  ij , (1.4) where A = [A ij ], B = [B ij ] are partitioned matrices of order m ×n and p ×q, respectively, A ij is of order m i ×n j , B kl of order p i ×q j , A ij ⊗B ij of order m i p i × n j q j (m =  t i =1 m i , n =  d j =1 n j , p =  t i =1 p i , q =  d j =1 q j ), and A ∗B of order M ×N (M =  t i =1 m i p i ,N =  d j =1 n j q j ). In general, AΘB = BΘA, A ⊗B = B ⊗A, A ∗B = B ∗A,butifA =[a ij ]isascalarmatrix and B = [B ij ] is a partitioned matrix, then A ∗ B = B ∗A. Additionally, Liu [5] shows that the Khatri-Rao product can be viewed as a generalized Hadamard product and the Tracy-Singh product as a generalized Kronecker product, as follows: (1) for a nonpartitioned matr i x A, their AΘB is A ⊗B, that is, AΘB =  a ij ΘB  ij =  a ij ⊗B kl  kl  ij =  a ij B kl  kl  ij =  a ij B  ij = A⊗B; (1.5) (2) for nonpartitioned matrices A and B of order m ×n, their A ∗B is A ◦B, that is, A ∗B =  a ij ⊗b ij  ij =  a ij b ij  ij = A◦B. (1.6) The Khatri-Rao and Tracy-Singh products are related by the following relation [5, 6]: A ∗B =Z T 1 (AΘB)Z 2 , (1.7) where A = [A ij ]ispartitionedwithA ij of order m i ×n j and B =[B kl ]ispartitionedwith B kl of order p k ×q l (m =  t i =1 m i , n =  d j =1 n j , p =  u k =1 p k , q =  v l =1 q l ), Z 1 is an mp× r (r =  t i =1 m i p i ) mat rix of zeros and ones, and Z 2 is an nq × s (s =  d j =1 n j q j )matrix Z. A. Al Zhour and A. Kilicman 3 of zeros and ones such that Z T 1 Z 1 = I r , Z T 2 Z 2 = I s (I r and I s are r × r and s ×s identity matrices, resp.). In particular, if m = n and p = q, then there exists an mp×r (r =  t i =1 m i p i )matrixZ such that Z T Z =I r (I r is an r ×r identity matrix) and A ∗B =Z T (AΘB)Z. (1.8) Here Z = ⎡ ⎢ ⎢ ⎣ Z 1 . . . Z t ⎤ ⎥ ⎥ ⎦ , (1.9) where each Z i = [ 0 i1 ··· 0 ii−1 I m i p i 0 ii+1 ··· 0 it ] T is an real matrix of zeros and ones, and 0 ik is a m i p i ×m i p k zero matrix for any k = i.NotealsothatZ T i Z i = I and Z T i  A ij ΘB  Z j = Z T i  A ij ⊗B kl  kl Z j = A ij ⊗B ij , i, j = 1,2, ,t. (1.10) In [5–8], the authors proved a number of equalities and inequalities involving Khatri- Rao and Hadamard products of two matrices. Here we extend these results in three ways. First, we establish new attractive equalities and inequalities involving Khatri-Rao prod- uct of matrices. Second, we recover and develop some results of Visick, for example, [8, Theorem 11, page 54]. This does not follow simply from the work of Visick. Third, the results are extended to the case of Khatri-Rao products of any finite number of matrices. This result leads to inequalities involving Hadamard product, as a special case. We use the following notations: (i) M m,n —the set of all m ×n matrices over the complex number field C and when m = n,wewriteM m instead of M m,n ; (ii) A T ,A ∗ ,A + ,A −1 —the tr anspose, conjugate t ranspose, Moore-Penrose inverse, and inverse of matrix A, respectively. For Hermitian mat rices A and B,therelationA>Bmeans that A −B>0 is a positive definite and the relation A ≥ B means A−B ≥0 is a positive semidefinite. Given a positive definite matrix A, its positive definite square root is denoted by A 1/2 . We use the known fact “for positive definite matrices A and B,therelationA ≥ B implies A 1/2 ≥ B 1/2 ” which is called the L ¨ owner-Heinz theorem. 2. Some notations and preliminary results Let A be a positive definite m ×m matrix. The spectral decomposition of matrix A assures that there exists a unitary matrix U such that A = U ∗ DU =U ∗ diag  λ i  U, U ∗ U =I m , (2.1) 4 Generalization inequalities for Khatri-Rao product where D = diag(λ i ) = diag(λ 1 , ,λ m ) is the diagonal matrix with diagonal entries λ i (λ i are the positive eigenvalues of A). For any real number r, A r is defined by A r = U ∗ D r U =U ∗ diag  λ r i  U. (2.2) If A ∈ M m,n is any matrix with rank (A) =s,thesingular value decomposition of A assures that there are unitary matrices U ∈ M m and V ∈ M n such that A = U  V ∗ . (2.3) Here  = [ W 0 00 ] ∈M m,n ,whereW = diag(σ 1 , ,σ s ) ∈M s is the diagonal matrix with di- agonal entries σ i (i = 1,2, ,s)andσ 1 ≥ σ 2 ≥···≥σ s > 0 are the singular values of A, that is, σ 1 ≥ σ 2 ≥···≥σ s > 0 are positive square roots of positive eigenvalues of A ∗ A and AA ∗ .TheMoore-Penrose inverse of A is defined by A + = V  W −1 0 00  U ∗ ∈ M n,m , (2.4) where W −1 = diag(σ −1 1 ,σ −1 2 , ,σ −1 s ) ∈ M s is the diagonal matrix with diagonal entries σ −1 i (i =1,2, ,s). A + is a unique matrix which satisfies the following conditions: AA + A =A, A + AA + = A + ,  AA +  ∗ = AA + ,  A + A  ∗ = A + A. (2.5) For any compatible partitioned matrices A, B, C,andD, we will make a frequent use of the following properties of the Tracy-Singh product (see e.g., [1, 3, 5, 10]): (a) (AΘB)(CΘD) = (AC)Θ(BD)ifAC and BD are well defined; (b) (AΘB) r = A r ΘB r if A ∈ M m , B ∈ M n are positive semidefinite matrices and r is any real number; (c) (AΘB) ∗ = A ∗ ΘB ∗ ; (d) (AΘB) + = A + ΘB + . If A ∈ M m and B ∈M n are positive semidefinite matrices, then (see, [3, 10]) (e) AΘB ≥ 0; (f) λ 1 (AΘB) =λ 1 (A)λ 1 (B), λ mn (AΘB) =λ m (A)λ n (B), where λ 1 (A), λ m (A) are the largest and smallest eigenvalues, respectively, of a matrix A, and λ 1 (B), λ n (B) are the largest and smallest eigenvalues, respectively, of a matr ix B. The Khatri-Rao and Tracy-Singh products of k matrices A i (1 ≤ i ≤k, k ≥2) will be denoted by  k i =1 ∗A i = A 1 ∗A 2 ∗···∗A k and  k i =1 ΘA i = A 1 ΘA 2 Θ ···ΘA k ,respec- tively. For a finite number of matrices A i (i = 1,2, ,k), the properties (a)–(d) become as in Lemma 2.1 and the connection between the Khatr i-Rao and Tracy-Singh products in (1.7)and(1.8)becomesasinLemma 2.2. Z. A. Al Zhour and A. Kilicman 5 Lemma 2.1. Let A i and B i (1 ≤i ≤k, k ≥2) be compatible partitioned matrices. Then (i)  k  i=1 ΘA i  k  i=1 ΘB i  =  k  i=1 Θ  A i B i   (2.6) if A i B i (1 ≤i ≤k, k ≥2) are well defined; (ii)  k  i=1 ΘA i  + = k  i=1 ΘA + i , k =2,3, ; (2.7) (iii)  k  i=1 ΘA i  ∗ = k  i=1 ΘA ∗ i ,  k  i=1 ∗A i  ∗ = k  i=1 ∗A ∗ i , k = 2,3, ; (2.8) (iv)  k  i=1 ΘA i  r = k  i=1 ΘA r i if A i ∈ M m(i) (1 ≤i ≤k, k ≥2) (2.9) are positive semidefinite matrices and r is any real number; (v)  k  i=1  A i ΘB i   =  k  i=1 A i  Θ  k  i=1 B i  , k =2,3, (2.10) Proof. The proof is immediately derived by induction on k.  Lemma 2.2. Let A i = [A (i) gh ] ∈ M m(i),n(i) (1 ≤ i ≤ k, k ≥ 2) be partitioned matrices with A (i) gh as the (g,h)th block submatrix (m =  k i =1 m(i), n =  k i =1 n(i), r =  t j =1  k i =1 m j (i), s =  t j =1  k i =1 n j (i), m(i) =  t j =1 m j (i), n(i) =  t j =1 n j (i)). Then there exist two real ma- trices Z 1 of order m ×r and Z 2 of order n ×s such that Z T 1 Z 1 = I r , Z T 2 Z 2 = I s (Z 1 , Z 2 are real matrices of zeros and ones) and k  i=1 ∗A i = Z T 1  k  i=1 ΘA i  Z 2 , k =2,3, , (2.11) where I r and I s are identity matrices of order r ×r and s ×s, respectively. In particular, if 6 Generalization inequalities for Khatri-Rao product m(i) = n(i)(1≤ i ≤k,k ≥ 2), then there exists an m ×r matrix Z of zeros and ones such that Z T Z =I r , k  i=1 ∗A i = Z T  k  i=1 ΘA i  Z, k = 2,3, , (2.12) and ZZ T is an m ×m diagonal matrix of zeros and ones, so 0 ≤ ZZ T ≤ I m , (2.13) where m =  k i =1 m(i). Proof. The special case in (2.12)ofLemma 2.2 is proved in [3, Corollary 2.2] and (2.13) follows immediately by the definition of matrix Z. We give proof of the general case in (2.11)ofLemma 2.2 for the sake of convenience. We proceed by induction on k.Ifk = 2, then (2.11)istrueby(1.7). Now suppose (2.11) holds for the Khatri-Rao product of k matrices, that is, there exist an m ×r matrix P kr of zeros and ones and an n ×s matr ix R ks of zeros and ones such that P T kr P kr = I r , R T ks R ks = I s ,and k  i=1 ∗A i = P T kr  k  i=1 ΘA i  R ks , k = 2,3, (2.14) WewillprovethatitistruefortheKhatri-Raoproductofk + 1 matrices. Then by (1.7), there exist an m(1)r ×r matrix Q 1 of zeros and ones and an n(1)s ×s matrix Q 2 of zeros and ones such that Q T 1 Q 1 = I r , Q T 2 Q 2 = I s ,and k+1  i=1 ∗A i = A 1 ∗  k+1  i=2 ∗A i  = Q T 1  A 1 Θ k+1  i=2 ∗A i  Q 2 = Q T 1  A 1 Θ  P T kr  k+1  i=2 ΘA i  R ks  Q 2 = Q T 1  I m(1) A 1 I n(1)  Θ  P T kr  k+1  i=2 ΘA i  R ks  Q 2 = Q T 1  I m(1) ΘP T kr   A 1 Θ  k+1  i=2 ΘA i   I n(1) ΘR ks  Q 2 = Q T 1  I m(1) ΘP T kr   k+1  i=1 ΘA i   I n(1) ΘR ks  Q 2 . (2.15) Letting Z 1 = (I m(1) ΘP kr )Q 1 and Z 2 = (I n(1) ΘR ks )Q 2 , the inductive step is complete. Here Q 1 = P 2r = P r , Q 1 = R 2s = R s , and it is a simple matter to verify that Z 1 =  I m(1) ΘP kr  P r = P (k+1)r , Z T 1 = P T r  I m(1) ΘP T kr  = P T (k+1)r , Z 2 =  I n(1) ΘR ks  R s = R (k+1)s , Z T 2 = R T s  I n(1) ΘR T ks  = R T (k+1)s . (2.16) Z. A. Al Zhour and A. Kilicman 7 Note that Z T 1 Z 1 = P T r  I m(1) ΘP T kr  I m(1) ΘP kr  P r = Q T 1  I m(1) ΘP T kr  I m(1) ΘP kr  Q 1 = Q T 1  I m(1) I m(1) ΘP T kr P kr  Q 1 = Q T 1  I m(1) ΘI r  Q 1  I m(1) ΘI r = I m(1)r  = Q T 1  I m(1)r  Q 1 = Q T 1 Q 1 = I r . (2.17) Similarly, it is easy to verify that Z T 2 Z 2 = I s .  Lemma 2.3. Let α be a nonempty subset of the set {1,2, ,m} and let A ∈M m be a positive semidefinite matrix. Then (see Chollet [4]) (i) if either −1 ≤r ≤0 or 1 ≤r ≤2, then A r (α) ≥A(α) r , ∀α; (2.18) (ii) if 0 ≤ r ≤ 1, then A r (α) ≤A(α) r , ∀α, (2.19) where A(α) is the principal submatrix of A whose entries are in the intersection of the rows and columns of A specified by α. Lemma 2.4. Let X j > 0(j = 1,2, ,k) be n × n matrices with eigenvalues in the interval [w,W] and U j ( j = 1,2, , k) are r ×m matricessuchthat  k j =1 U j U ∗ j = I. Then (see Mond and Pe ˇ cari ´ c[7]) (i) for every real p>1 and p<0, k  j=1 U j X p j U ∗ j ≤ μ  k  j=1 U j X j U ∗ j  p , (2.20) where μ = δ p −δ (p −1)(δ −1)  p −1 p δ p −1 δ p −δ  p , δ = W w . (2.21) While for 0 <p<1, the reverse inequality holds in (2.20); (ii) for every real p>1 and p<0,  k  j=1 U j X p j U ∗ j  −  k  j=1 U j X j U ∗ j  p ≤ γ{I}, (2.22) where γ = Ww p −wW p W −w +(p −1)  1 p W p −w P W −w  p/(p−1) . (2.23) While for 0 <p<1 , the reverse inequality holds in (2.22). 8 Generalization inequalities for Khatri-Rao product 3. New applications and results Based on the basic results in Section 2 and the general connection between the Khatri-Rao and Tracy-Singh products in Lemma 2.2, we generalize and derive some equalities and inequalities in works of Visick [8, Corollary 3, Theorem 4], Chollet [4], and Mond and Pe ˇ cari ´ c[7] with respect to the Khatri-Rao product and extend these results to any finite number of matrices. These results lead to inequalities involving Hadamard products, as a special case. Theorem 3.1. Let A i = [A (i) gh ] ∈ M m(i),n(i) (1 ≤ i ≤k, k ≥2) be partitioned matrices with A (i) gh as the (g,h)th block submatrix (m =  k i =1 m(i), n =  k i =1 n(i)) and let Z 1 and Z 2 be the real matrices of zeros and ones that satisfy (2.11). Then (i) there exists an m ×(m −r) matrix Q (m) of zeros and ones such that the block matrix Ω = [ Z 1 Q (m) ] is an m×m permutat ion matrix. Q (m) is not unique but for any such choice of Q (m) , Z T 1 Q (m) = 0, Q T (m) Q (m) = I m−r , Q (m) Q T (m) + Z 1 Z T 1 = I m (3.1) (ii) for any m ×n matrix L, Z T 1 LL ∗ Z 1 ≥  Z T 1 LZ 2  Z T 1 LZ 2  ∗ ≥ 0. (3.2) Proof. Though the proof is quite similar to the proof of [8, Corollary 3(iii) and (vii)] for Hadamard product, we give proof for the sake of convenience. (i) It is evident from the structure of Z 1 that it may be considered as part of an m×m permutation matrix Ω = [ Z 1 Q (m) ], where Q (m) is an m ×(m −r) matrix of zeros and ones. For example, when k = 2, then Q (2) is not unique (see, [8, page 49]). Using the properties of a permutation matrix together with the definition of Ω = [ Z 1 Q (m) ], we have I m = ΩΩ T =  Z 1 Q (m)   Z T 1 Q T (m)  = Q (m) Q T (m) + Z 1 Z T 1 , I m =  I r 0 0 I m−r  = Ω T Ω =  Z T 1 Q T m   Z 1 Q (m)  =  Z T 1 Z 1 Z T 1 Q (m) Q T (m) Z 1 Q T (m) Q (m)  . (3.3) From these come the required results in (i), that is, Z T 1 Q (m) = 0, Q T (m) Q (m) = I m−r , Q (m) Q T (m) + Z 1 Z T 1 = I m . (3.4) (ii) By (2.13)ofLemma 2.2,wehaveI n ≥ Z 2 Z T 2 ≥ 0andso Z T 1 LL ∗ Z 1 ≥ Z T 1 LZ 2 Z T 2 L ∗ Z 1 =  Z T 1 LZ 2  Z T 1 LZ 2  ∗ ≥ 0. (3.5) We now generalize [8, Theorem 4] to the case of Khatri-Rao product involving a finite number of matrices.  Z. A. Al Zhour and A. Kilicman 9 Theorem 3.2. Let A i = [A (i) gh ] ∈M m,n (1 ≤i ≤ k, k ≥2) be partitioned matrices with A (i) gh as the (g,h)th block submatrix. Let Z 1 be an m k ×r matrix of zeros and ones that satisfies (2.12)andletQ (n) be an n k ×(n k −s) matrix of zeros and ones that satisfies (3.1). Then k  i=1 ∗(A i A ∗ i ) =  k  i=1 ∗(A i )  k  i=1 ∗A i  ∗ + Z T 1  k  i=1 ΘA i  Q (n) Q T (n)  k  i=1 ΘA i  ∗ Z 1 =  k  i=1 ∗(A i )  k  i=1 ∗A i  ∗ +  Z T 1  k  i=1 ΘA i  Q (n)  Z T 1  k  i=1 ΘA i  Q (n)  ∗ , (3.6) and hence k  i=1 ∗  A i A ∗ i  ≥  k  i=1 ∗(A i )  k  i=1 ∗A i  ∗ , k =2,3, (3.7) Proof. From Lemma 2.1(i) and (iii), we have k  i=1 Θ  A i A ∗ i  =  k  i=1 ΘA i  k  i=1 ΘA i  ∗ . (3.8) But by Theorem 3.1(i), there exist an n k ×s matr ix Z 2 of zeros and ones that satisfies (2.12)andann k ×(n k −s)matrixQ (n) of zeros and ones that satisfies (3.1)suchthat Z 2 Z T 2 + Q (n) Q T (n) = I n k and k  i=1 Θ  A i A ∗ i  =  k  i=1 ΘA i   Z 2 Z T 2 + Q (n) Q T (n)   k  i=1 ΘA i  ∗ =  k  i=1 ΘA i   Z 2 Z T 2   k  i=1 ΘA i  ∗ +  k  i=1 ΘA i   Q (n) Q T (n)   k  i=1 ΘA i  ∗ . (3.9) Since A i (1 ≤i ≤ k, k ≥ 2) are rectangular partitioned matrices of order m ×n,thendue to (2.11)ofLemma 2.2 there exist two real matrices Z 1 and Z 2 of zeros and ones of order m k ×r and n k ×s, respectively, such that k  i=1 ∗A i = Z T 1  k  i=1 ΘA i  Z 2 , k = 2,3, (3.10) But because A i A ∗ i (1 ≤ i ≤ k, k ≥ 2) are square matrices of order m ×m,thendueto (2.12)ofLemma 2.2 there exists a real matrix Z 1 of zeros and ones of order m k ×r such that k  i=1 ∗  A i A ∗ i  = Z T 1  k  i=1 Θ  A i A ∗ i   Z 1 , k = 2,3, (3.11) 10 Generalization inequalities for Khatri-Rao product Due to (3.9), (3.10), and (3.11), we have k  i=1 ∗  A i A ∗ i  = Z T 1  k  i=1 Θ  A i A ∗ i   Z 1 = Z T 1  k  i=1 ΘA i  Z 2 Z T 2  k  i=1 ΘA i  ∗ Z 1 + Z T 1  k  i=1 ΘA i   Q (n) Q T (n)   k  i=1 ΘA i  ∗ Z 1 =  Z T 1  k  i=1 ΘA i  Z 2  Z T 1  k  i=1 ΘA i  Z 2  ∗ + Z T 1  k  i=1 ΘA i   Q (n) Q T (n)   k  i=1 ΘA i  ∗ Z 1 =  k  i=1 ∗  A i   k  i=1 ∗A i  ∗ + Z T 1  k  i=1 ΘA i  Q (n) Q T (n)  k  i=1 ΘA i  ∗ Z 1 =  k  i=1 ∗  A i   k  i=1 ∗A i  ∗ +  Z T 1  k  i=1 ΘA i  Q (n)  Z T 1  k  i=1 ΘA i  Q (n)  ∗ . (3.12)  If we put k = 2inTheorem 3.2, we obtain the following corollary. Corollary 3.3. Let A i = [A (i) gh ] ∈M m,n (1 ≤i ≤2) be part itioned matrices with A (i) gh as the (g,h)th block submatrix. Let Z 1 be an m 2 ×r matrix of zeros and ones that satisfies (1.8) and let Q (n) be an n 2 ×(n 2 −s) matrix of zeros and ones that satisfies (3.1). Then A 1 A ∗ 1 ∗A 2 A ∗ 2 =  A 1 ∗A 2  A 1 ∗A 2  ∗ + Z T 1  A 1 ΘA 2  Q (n) Q T (n)  A 1 ΘA 2  ∗ Z 1 , (3.13) and hence A 1 A ∗ 1 ∗A 2 A ∗ 2 ≥  A 1 ∗A 2  A 1 ∗A 2  ∗ . (3.14) Corollary 3.4. Let A i = [A (i) gh ] ∈M m,n (1 ≤i ≤k, k ≥ 2) be partitioned matrices with A (i) gh as the (g,h)th block submatrix. Let Z 1 be an m k ×r matrix of zeros and ones that satisfies (2.12)andletQ (n) be an n k ×(n k −s) matrix of zeros and ones that satisfies (3.1). Then the following statements are equivalent: (i) k  i=1 ∗  A i A ∗ i  =  k  i=1 ∗  A i   k  i=1 ∗A i  ∗ , k = 2,3, ; (3.15) (ii) Z T 1  k  i=1 ΘA i  Q (n) = 0, k =2,3, ; (3.16) [...]... obtained in Sections 3 and 4 are quite general These results lead to inequalities involving Hadamard product, as a special case, for nonpartitioned matrices Ai (i = 1,2, ,k, k ≥ 2) with the Hadamard product and Kronecker product replacing the Khatri-Rao product and Tracy-Singh product, respectively Now we utilize the commutativity of the Hadamard product to develop, for instance, (3.7) of Theorem 3.2 This... It is also known that if matrix A is square A B and nonsingular, then A+ = A−1 and [ B∗ D ] ≥ 0 if and only if D ≥ B∗ A−1 B Let Z1 and Z2 be the real matrices of zeros and ones of order m × r and n × s, respectively, that satisfy (2.11) in Lemma 2.2 Now another way to use Lemma 2.2 to generate inequalities involving the Khatri-Rao product is by using the following obvious inequality: TT ∗ = T1 T2 ∗... A∗−1 j j+1 j + i= j Now the application of (4.36) and the commutativity of the Hadamard product yield T Pkm LL∗ Pkm = α2 + · · · + α2 1 k where μr = k w αw α(w+r) k i =1 ◦ Ai A∗ i k −1 k μr + r =1 w=1 ◦ Aw A∗ (w+r) and w + r ≡ (w + r) mod k with 1 ≤ (w + r) ≤ k , (4.40) 20 Generalization inequalities for Khatri-Rao product Also by (4.36) and the commutativity of the Hadamard product, we obtain T T Pkm... 1–3, 267–277 , Several inequalities involving Khatri-Rao products of positive semidefinite matrices, Lin[6] ear Algebra and Its Applications 354 (2002), no 1–3, 175–186 [7] B Mond and J E Peˇ ari´ , Matrix inequalities for convex functions, Journal of Mathematical Analc c ysis and Applications 209 (1997), no 1, 147–153 [8] G Visick, A quantitative version of the observation that the Hadamard product is... submatrix of the Kronecker product, Linear Algebra and Its Applications 304 (2000), no 1–3, 45–68 [9] F Zhang, Matrix Theory Basic Results and Techniques, Universitext, Springer, New York, 1999 [10] X Zhang, Z.-P Yang, and C.-G Cao, Inequalities involving Khatri-Rao products of positive semidefinite matrices, Applied Mathematics E-Notes 2 (2002), 117–124 Zeyad Abdel Aziz Al Zhour: Department of Mathematics and. .. (UPM) under the Grant IRPA09-02-04-0259-EA001 References [1] Z A Al Zhour and A Kilicman, New Holder-type inequalities for the Tracy-Singh and KhatriRao products of positive matrices, Proceedings of the International Conference on Mathematics, Statistics and Their Applications, vol 1, North Sumatera, 2005, pp 1–7 [2] A Albert, Conditions for positive and nonnegative definiteness in terms of pseudoinverses,... Applied Mathematics 17 (1969), no 2, 434–440 [3] C.-G Cao, X Zhang, and Z.-P Yang, Some inequalities for the Khatri-Rao product of matrices, Electronic Journal of Linear Algebra 9 (2002), 276–281 [4] J Chollet, Some inequalities for principal submatrices, The American Mathematical Monthly 104 (1997), no 7, 609–617 [5] S Liu, Matrix results on the Khatri-Rao and Tracy-Singh products, Linear Algebra and Its... , T2 , and L are possible which lead to quite different inequalities involving Khatri-Rao products However, there exist some inequalities that do not seem to follow directly from (1.7) or (2.11), but follow easily from (4.4) and (3.2) Based on (4.4) and (3.2) we generalize some inequalities in works of Visick [8, Corollary 13, Remark in page 56, Theorems 11, 17, and 20] and establish some new inequalities. .. (4.35) Z A Al Zhour and A Kilicman 19 We will extend this inequality to the case of products involving any finite number of matrices If the Tracy-Singh and Khatri-Rao products are replaced by the Kronecker and Hadamard products in Lemma 2.2, respectively, we obtain the following corollary Corollary 4.14 Let Ai ∈ Mm,n (1 ≤ i ≤ k, k ≥ 2) Then k k i=1 T ◦Ai = Pkm ⊗Ai Pkn , (4.36) i =1 (m) (m) (m) where Pkm... obtain i (ii) Remark 3.7 It is easy to give another proof of Theorem 3.6 by replacing A by in Lemma 2.3 and applying (2.12) of Lemma 2.2 k i=1 ΘAi Theorem 3.8 Let Ai > 0 be compatible partitioned matrices such that k=1 ΘAi > 0 (1 ≤ i i ≤ k, k ≥ 2) Let W and w be the largest and smallest eigenvalues of k=1 ΘAi , respectively i Then (i) for every real p > 1 and p < 0, k p k p ∗Ai ≤ μ i=1 ∗Ai , k = 2,3, . (3.2) Proof. Though the proof is quite similar to the proof of [8, Corollary 3(iii) and (vii)] for Hadamard product, we give proof for the sake of convenience. (i) It is evident from the structure of. page 56, Theorems 11, 17, and 20] and establish some new inequalities involving Khatri-Rao products of several positive matrices. Theorem 4.1. Let A 1 and A 2 be compatible partitioned matrices. Then A 1 A ∗ 1 ∗A 2 A ∗ 2 +. recover and develop some results of Visick. Third, the results are extended to the case of Khatri- Rao product of any finite number of matrices. These results lead to inequalities involving Hadamard product,