Journal of Inequalities in Pure and Applied Mathematics

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1 Journal of Inequalities in Pure and Applied Mathematics Volume 7, Issue 1, Article 34, 2006 MATRIX EQUALITIES AND INEQUALITIES INVOLVING KHATRI-RAO AND TRACY-SINGH SUMS ZEYAD AL ZHOUR AND ADEM KILICMAN DEPARTMENT OF MATHEMATICS AND INSTITUTE FOR MATHEMATICAL RESEARCH UNIVERSITY PUTRA MALAYSIA (UPM), MALAYSIA 43400, SERDANG, SELANGOR, MALAYSIA Received 05 August, 2004; accepted 02 September, 2005 Communicated by R.P. Agarwal ABSTRACT. The Khatri-Rao and Tracy-Singh products for partitioned matrices are viewed as generalized Hadamard and generalized Kronecker products, respectively. We define the Khatri- Rao and Tracy-Singh sums for partitioned matrices as generalized Hadamard and generalized Kronecker sums and derive some results including matrix equalities and inequalities involving the two sums. Based on the connection between the Khatri-Rao and Tracy-Singh products (sums) and use mainly Liu s, Mond and Pečarić s methods to establish new inequalities involving the Khatri-Rao product (sum). The results lead to inequalities involving Hadamard and Kronecker products (sums), as a special case. Key words and phrases: Kronecker product (sum), Hadamard product (sum), Khatri-Rao product (sum), Tracy-Singh product (sum), Positive (semi)definite matrix, Unitarily invariant norm, Spectral norm, P-norm, Moore- Penrose inverse Mathematics Subject Classification. 15A45; 15A INTRODUCTION The Hadamard and Kronecker products are studied and applied widely in matrix theory, statistics, econometrics and many other subjects. Partitioned matrices are often encountered in statistical applications. For partitioned matrices, The Khatri-Rao product viewed as a generalized Hadamard product, is discussed and used in [7, 6, 14] and the Tracy-Singh product, as a generalized Kronecker product, is discussed and applied in [7, 5, 12]. Most results provided are equalities associated with the products. Rao, Kleffe and Liu in [13, 8] presented several matrix inequalities involving the Khatri-Rao product, which seem to be most existing results. In [7], Liu established the connection between Khatri-Rao and Tracy-Singh products based on two selection matrices Z 1 and Z 2. This connection play an important role to give inequalities involving the two products ISSN (electronic): c 2006 Victoria University. All rights reserved

2 2 ZEYAD AL ZHOUR AND ADEM KILICMAN with statistical applications. In [10], Mond and Pečarić presented matrix versions, with matrix weights. In [2, (2004)], Hiai and Zhan proved the following inequalities: (*) AB A B A B A B A + B A + B, A + B A + B for any invariant norm with diag(1, 0,..., 0) 1 and A, B are nonzero positive definite matrices. In the present paper, we make a further study of the Khatri-Rao and Tracy-Singh products. We define the Khatri-Rao and Tracy-Singh sums for partitioned matrices and use mainly Liu s, Mond and Pečarić s methods to obtain new inequalities involving these products (sums).we collect several known inequalities which are derived as a special cases of some results obtained. We generalize the inequalities in Eq (*) involving the Hadamard product (sum) and the Kronecker product (sum). 2. BASIC DEFINITIONS AND RESULTS 2.1. Basic Definitions on Matrix Products. We introduce the definitions of five known matrix products for non-partitioned and partitioned matrices. These matrix products are defined as follows: Definition 2.1. Consider matrices A = (a ij ) and C = (c ij ) of order m n and B = (b kl ) of order p q. The Kronecker and Hadamard products are defined as follows: (1) Kronecker product: (2.1) A B = (a ij B) ij, where a ij B is the ij th submatrix of order p q and A B of order mp nq. (2) Hadamard product: (2.2) A C = (a ij c ij ) ij, where a ij c ij is the ij th scalar element and A C is of order m n. Definition 2.2. Consider matrices A = (a ij ) and B = (b kl ) of order m m and n n respectively. The Kronecker sum is defined as follows: (2.3) A B = A I n + I m B, where I n and I m are identity matrices of order n n and m m respectively, and A B of order mn mn. Definition 2.3. Consider matrices A and C of order m n, and B of order p q. Let A = (A ij ) be partitioned with A ij of order m i n j as the ij th submatrix,c = (C ij ) be partitioned with C ij of order m i n j as the ij th submatrix, and B = (B kl ) be partitioned with B kl of order p k q l as the kl th submatrix, where,m = r i=1 m i, n = s j=1 n j, p = t k=1 p k, q = h l=1 q l are partitions of positive integers m, n, p, and q. The Tracy-Singh and Khatri-Rao products are defined as follows: (1) Tracy-Singh product: (2.4) AΠB = (A ij ΠB) ij = ( (A ij B kl ) kl )ij,

3 MATRIX EQUALITIES AND INEQUALITIES INVOLVING KHATRI-RAO AND TRACY-SINGH SUMS 3 where A ij is the ij th submatrix of order m i n j, B kl is the kl th submatrix of order p k q l, A ij ΠB is the ij th submatrix of order m i p n j q, A ij B kl is the kl th submatrix of order m i p k n j q l and AΠB of order mp nq. Note that (i) For a non partitioned matrix A, their AΠB is A B, i.e., for A = (a ij ), where a ij is scalar, we have, AΠB = (a ij ΠB) ij = ( (a ij B kl ) kl )ij = ( (a ij B kl ) kl )ij = (a ijb) ij = A B. (ii) For column wise partitioned A and B, their AΠB is A B. (2) Khatri-Rao product: (2.5) A B = (A ij B ij ) ij, where A ij is the ij th submatrix of order m i n j, B ij is the ij th submatrix of order ( p i q j, A ij B ij is the ij th submatrix of order m i p i n j q j and A B of order M N M = r i=1 m ip i, N = s j=1 n jq j ). Note that (i) For a non partitioned matrix A, their A B is A B, i.e., for A = (a ij ),where a ij is scalar, we have, A B = (a ij B ij ) ij = (a ij B) ij = A B. (ii) For non partitioned matrices A and B, their A B is A B, i.e., for A = (a ij ) and B = (b ij ), where a ij and b ij are scalars, we have, A B = (a ij b ij ) ij = (a ij b ij ) ij = A B Basic Connections and Results on Matrix Products. We introduce the connection between the Katri-Rao and Tracy-Singh products and the connection between the Kronecker and Hadamard products, as a special case, which are important in creating inequalities involving these products. We write A B in the Löwner ordering sense that A B 0 is positive semi-definite, for symmetric matrices A and B of the same order and A + and A indicate the Moore-Penrose inverse and the conjugate of the matrix A, respectively. Lemma 2.1. Let A = (a ij ) and B = (b ij ) be two scalar matrices of order m n. Then (see [15]) (2.6) A B = K 1(A B)K 2 where K 1 and K 2 are two selection matrices of order n 2 n and m 2 m, respectively, such that K 1K 1 = I m and K 2K 2 = I n. In particular, for m = n, we have K 1 = K 2 = K and (2.7) A B = K (A B)K Lemma 2.2. Let A and B be compatibly partitioned. Then (see [8, p ] and [7, p. 272]) (2.8) A B = Z 1 (AΠB) Z 2, where Z 1 and Z 2 are two selection matrices of zeros and ones such that Z 1Z 1 = I 1 and Z 2Z 2 = I 2, where I 1 and I 2 are identity matrices.

4 4 ZEYAD AL ZHOUR AND ADEM KILICMAN In particular, when A and B are square compatibly partitioned matrices, then we have Z 1 = Z 2 = Z such that Z Z = I and (2.9) A B = Z (AΠB) Z. Note that, for non-partitioned matrices A, B, Z 1 and Z 2, Lemma 2.2 leads to Lemma 2.1, as a special case. Lemma 2.3. Let A, B, C, D and F be compatibly partitioned matrices. Then (2.10) (2.11) (2.12) (2.13) (2.14) (2.15) (2.16) (2.17) (2.18) (2.19) Proof. Straightforward. (AΠB)(CΠD) = (AC)Π(BD) (AΠB) + = A + ΠB + (A + C)Π(B + D) = AΠB + AΠD + CΠB + CΠD (AΠB) = A ΠB AΠB BΠA A B B A in general in general B F = F B where F = (f ij ) and f ij is a scalar (A B) = A B (A + C) (B + D) = A B + A D + C B + C D (A B)Π(C D) = (AΠC) (BΠD) Lemma 2.4. Let A and B be compatibly partitioned matrices. Then (2.20) (AΠB) r = A r ΠB r, for any positive integer r. Proof. The proof is by induction on r and using Eq. (2.10). Theorem 2.5. Let A 0 and B 0 be compatibly partitioned matrices. Then (2.21) (AΠB) α = A α ΠB α for any positive real α. Proof. By using Eq (2.20), we have AΠB = (A 1/n ΠB 1/n ) n, for any positive integer n. So it follows that (AΠB) 1/n = A 1/n ΠB 1/n. Now (AΠB) m/n = A m/n ΠB m/n, for any positive integers n, m. The Eq (2.21) now follows by a continuity argument. Corollary 2.6. Let A and B be compatibly partitioned matrices. Then (2.22) AΠB = A Π B, where A = (A A) 1/2 Proof. Applying Eq (2.10) and Eq (2.21), we get the result. Theorem 2.7. Let A = (A ij ) and B = (B kl ) be partitioned matrices of order m m, and n n respectively, where m = r i=1 m i, n = t k=1 n k.then (2.23) (2.24) (a) tr(aπb) = tr(a) tr(b) (b) AΠB p = A p B p, where A p = [tr A p ] 1/p, for all 1 p <. Proof. (a) Straightforward. (b) Applying Eq (2.22) and Eq (2.23), we get the result.

5 MATRIX EQUALITIES AND INEQUALITIES INVOLVING KHATRI-RAO AND TRACY-SINGH SUMS 5 Theorem 2.8. Let A, B and I be compatibly partitioned matrices. Then (2.25) (AΠI)(IΠB) = (IΠB)(AΠI) = AΠB. If f(a) is an analytic function on a region containing the eigenvalues of A, then (2.26) f(iπa) = IΠf(A) and f(aπi) = f(a)πi Proof. The proof of Equation (2.25) is straightforward on applying Eq (2.10). Equation (2.26) can be proved as follows: Since f(a)is an analytic function, then f(a) = k=0 α ka k. Applying Eq (2.10) we get: f(iπa) = α k (IΠA) k = α k (IΠA k ) = IΠ α k A k = IΠf(A). k=0 Corollary 2.9. Let A, B and I be compatibly partitioned matrices. Then (2.27) e AΠI = e A ΠI and e IΠA = IΠe A. k=0 Lemma Let H 0 be a n n matrix with nonzero eigenvalues λ 1 λ k (k n) and X be a m m matrix such that X = H 0 X, where H 0 = HH +. Then (see [6, Section 2.3]) (2.28) (X HX) + X + H + X + (λ 1 + λ k ) 2 k=0 (4λ 1 λ k ) (X HX) +. Theorem Let A 0 and B 0 be compatibly partitioned matrices such that A 0 = AA + and B 0 = BB +. Then (see [8, Section 3]) (2.29) (A B 0 + A 0 B)(A B) + (A B 0 + A 0 B) A B + + A + B + 2A 0 B 0 Theorem Let A > 0 and B > 0 be n n compatibly partitioned matrices with eigenvalues contained in the interval between m and M (M m). Let I be a compatible identity matrix. Then (see [8, Section 3]). (2.30) A B 1 + A 1 B m2 + M On the Tracy-Singh Sum. mm I and A A 1 m2 + M 2 2mM 3. MAIN RESULTS Definition 3.1. Consider matrices A and B of order m m and n n respectively. Let A = (A ij ) be partitioned with A ij of order m i m i as the ij th submatrix, and let B = (B ij ) be partitioned with B ij of order n k n k as the ij th submatrix ( m = r i=1 m i, n = t k=1 n k). The Tracy-Singh sum is defined as follows: (3.1) A B = AΠI n + I m ΠB, where I n = I n1 +n 2 + +n t = blockdiag(i n1, I n2,..., I nt ) is an n n identity matrix, I m = I m1 +m 2 + +m r = blockdiag(i m1, I m2,..., I mr ) is an m m identity matrix, I nk is an n k n k identity matrix (k = 1,..., t), I mi is an m i m i identity matrix (i = 1,..., r) and A B is of order mn mn. Note that for non-partitioned matrices A and B, their A B is A B. Theorem 3.1. Let A 0, B 0, C 0 and D 0 be compatibly partitioned matrices. Then (3.2) (A B)(C D) AC BD. I

6 6 ZEYAD AL ZHOUR AND ADEM KILICMAN Proof. Applying Eq (3.1) and Eq (2.10), we have (A B)(C D) = (AΠI + IΠB)(CΠI + IΠD) = (AΠI)(CΠI) + (AΠI)(IΠD) + (IΠB)(CΠI) + (IΠB)(IΠD) = ACΠI + AΠD + CΠB + IΠBD = AC BD + AΠD + CΠB AC BD. In special cases of Eq (3.2), if C = A, D = B, we have (3.3) (A B)(A B) AA BB and if C = A, D = B, we have (3.4) (A B) 2 A 2 B 2. More generally, it is easy by induction on w we can show that ifa 0 and B 0 are compatibly partitioned matrices. Then w 1 ( ) w (3.5) (A B) w = A w B w + (A w k ΠB k ); k k=1 (3.6) (A B) w A w B w for any positive integer w. Theorem 3.2. Let A and B be partitioned matrices of order m m and n n, respectively, ( m = r i=1 m i, n = t k=1 n k). Then (3.7) tr(a B) = n tr(a) + m tr(b), (3.8) A B p p n A p + p m B p, where A p = [tr A p ] 1/p, 1 p <, and (3.9) e A B = e A Πe B. Proof. For the first part, on applying Eq (2.23), we obtain To prove (3.8), we apply Eq (2.24), to get tr(a B) = tr [(AΠI n ) + (I m ΠB)] = tr(aπi n ) + tr(i m ΠB) = tr(a) tr(i n ) + tr(i m ) tr(b) = n tr(a) + m tr(b). A B p = (AΠI n ) + (I m ΠB) p AΠI n p + I m ΠB p = A p I n p + I m p B p = p n A p + p m B p. For the last part, applying Eq (2.25), Eq (2.27) and Eq (2.10), we have e A B = e (AΠIn)+(ImΠB) = e (AΠIn) e (ImΠB) = (e A ΠI n )(I m Πe B ) = e A Πe B.

7 MATRIX EQUALITIES AND INEQUALITIES INVOLVING KHATRI-RAO AND TRACY-SINGH SUMS 7 Theorem 3.3. Let A and Bbe non singular partitioned matrices of order m m and n n respectively, (m = r i=1 m i, n = t k=1 n k).then (3.10) (i) (A B) 1 = (A 1 B 1 ) 1 (A 1 ΠB 1 ) (3.11) (3.12) (ii) (A B) 1 = (A 1 ΠI n )(A 1 B 1 ) 1 (I m ΠB 1 ) (iii) (A B) 1 = (I m ΠB 1 )(A 1 B 1 ) 1 (A 1 ΠI n ) Proof. (i) Applying Eq (2.10), we have (A B) 1 = [I m ΠB + AΠI n ] 1 Similarly, we obtain (ii) and (iii). = [(I m ΠB)(I m ΠI n ) + (I m ΠB)(AΠB 1 )] 1 = [(I m ΠB)(I m ΠI n + AΠB 1 )] 1 = [(I m ΠI n + AΠB 1 )] 1 [I m ΠB] 1 = [(AΠI n )(A 1 ΠI n ) + (AΠI n )(I m ΠB 1 )] 1 [I m ΠB 1 ] = [(AΠI n ){A 1 ΠI n + I m ΠB 1 }] 1 [I m ΠB 1 ] = [(AΠI n )(A 1 B 1 )] 1 [I m ΠB 1 ] = (A 1 B 1 ) 1 (A 1 ΠI n )(I m ΠB 1 ) = (A 1 B 1 ) 1 (A 1 ΠB 1 ). Theorem 3.4. Let A 0 and I be compatibly partitioned matrices such that A + ΠI = IΠA +. Then (3.13) A A + 2AA + ΠI. Proof. We know that A I = AΠI + IΠI > AΠI. Denote H = MΠI 0. By virtue of H + H + 2HH + and Eq (2.10), we have Since, A + ΠI = IΠA +, we get the result On the Khatri-Rao Sum. AΠI + (AΠI) + 2(AΠI)(AΠI) + = 2AA + ΠI Definition 3.2. Let A, B, I n and I m be partitioned as in Definition 3.1. Then the Khatri-Rao sum is defined as follows: (3.14) A B = A I n + I m B Note that, for non-partitioned matrices A and B, their A B is A B, and for non-partitioned matrices A, B, I n and I m, their A B is A B (Hadamard sum, see Definition 4.1, Eq(4.1), Section 4). Theorem 3.5. Let A and B be compatibly partitioned matrices. Then (3.15) A B = Z (A B)Z, where Z is a selection matrix as in Lemma 2.2. Proof. Applying Eq (2.9), we have A I = Z (AΠI)Z, I B = Z (IΠB)Z and A B = A I + I B = Z (AΠI) Z + Z (IΠB) Z = Z (A B)Z.

8 8 ZEYAD AL ZHOUR AND ADEM KILICMAN Corollary 3.6. Let A 0 and I be compatibly partitioned matrices such that A + ΠI = IΠA +. Then (3.16) A A + 2AA + I Proof. Applying Eq(3.13) and Eq (3.15), we get the result. Corollary 3.7. Let A > 0 be compatibly partitioned with eigenvalues contained in the interval between m and M (M m). Let I be a compatible identity matrix such that A 1 I = I A 1. Then (3.17) A A 1 m2 + M 2 mm I. Proof. Applying Eq (2.30) and taking B = I, we get the result. Corollary 3.8. Let A 0 and I be compatibly partitioned, where A 0 = AA + such that A 0 I = I A 0. Then (3.18) (A A 0 )(A I) + (A A 0 ) A I + A + I + 2A 0 I and if A + I = I A +, we have (3.19) (A A 0 )(A I) + (A A 0 ) A A + + 2A 0 I. Proof. Applying Eq (2.29) and taking B = I, we get the results. Mond and Pečarić (see [10]) proved the following result: If X j (j = 1, 2,..., k) are positive definite Hermitian matrices of order n n with eigenvalues in the interval [m, M] and U j (j = 1, 2,..., k) are r n matrices such that k j=1 U juj = I. Then (a) For p < 0 or p > 1, we have ( k k ) p (3.20) U j X p j U j λ U j X j Uj where, (3.21) λ = j=1 j=1 { } γ p γ p(γ γ p p ), γ = M (p 1)(γ 1) (1 p)(γ p 1) m. While, for 0 < p < 1, we have the reverse inequality in Eq (3.20). (b) For p < 0 or p > 1, we have ( k ) ( k ) p (3.22) U j X p j U j U j X j Uj αi, j=1 where, { } M (3.23) α = m p p m p p [ p 1 { } ] M p m p M p m p 1 p 1 + m. p(m m) (M m) p(m m) While, for 0 < p < 1, we have the reverse inequality in Eq (3.22). We have an application to the Khatri-Rao product and Khatri-Rao sum. Theorem 3.9. Let A and B be positive definite Hermitian compatibly partitioned matrices and let m and M be, respectively, the smallest and the largest eigenvalues of AΠB. Then j=1

9 MATRIX EQUALITIES AND INEQUALITIES INVOLVING KHATRI-RAO AND TRACY-SINGH SUMS 9 (a) For p a nonzero integer, we have (3.24) A p B p λ(a B) p where, λ is given by Eq (3.21). While, for 0 < p < 1, we have the reverse inequality in Eq (3.24). (b) For p a nonzero integer, we have (3.25) (A p B p ) (A B) p αi, where α is given by Eq (3.23). While, for 0 < p < 1, we have the reverse inequality in Eq (3.25). Proof. In Eq (3.20) and Eq (3.22), take k = 1 and instead of U, use Z, the selection matrix which satisfy the following property: A B = Z (AΠB)Z, Z Z = I. Making use of the fact in Eq (2.21) that for any real n (positive or negative), we have (AΠB) n = A n ΠB n, then, with Z, AΠB, Z substituted for U, X, U, we have from Eq (3.20) A p B p = Z (A p B p )Z = Z (A B) p Z where, λ is given by Eq (3.21) Similarly, from Eq (3.22), we obtain for where, α is given by Eq (3.23). Special cases include from Eq (3.24): λ {Z (AΠB)Z} p = λ(a B) p, (A p B p ) (A B) p αi (3.26) A 2 B 2 (M + m)2 4Mm {A B}2 (3.27) A 1 B 1 (M + m)2 {A B} 1 4Mm Similarly, special cases include from Eq (3.25): (3.28) (A 2 B 2 ) (A B) (M m)2 I M m (3.29) (A 1 B 1 ) (A B) 1 Mm {I}, where results in Eq (3.26), Eq (3.27), and Eq (3.28) are given in [7]. Theorem Let A and B be positive definite Hermitian compatibly partitioned matrices. Let m 1 and M 1 be, respectively, the smallest and the largest eigenvalues of AΠI and m 2 and M 2, respectively, the smallest and the largest eigenvalues of IΠB. Then

10 10 ZEYAD AL ZHOUR AND ADEM KILICMAN (a) For p a nonzero integer, we have (3.30) A p B p max {λ 1, λ 2 } (A B) p where, (3.31) λ 1 = (γ p { 1 γ 1 ) p(γ1 γ p } p 1) [(p 1)(γ 1 1)] [(1 p)(γ p, γ 1 = M 1, 1 1)] m 1 (3.32) λ 2 = (γ p { 2 γ 2 ) p(γ2 γ p } p 2) [(p 1)(γ 2 1)] [(1 p)(γ p, γ 2 = M )] m 2 While, for 0 < p < 1, we have the reverse inequality in Eq (3.30). (b) For p a nonzero integer, we have (3.33) (A p B p ) (A B) p max {α 1, α 2 } I where, (3.34) α 1 = m p 1 (3.35) α 2 = m p 2 { M p 1 m p } p { p 1 { 1 M p + 1 m p 1 M p 1 m p } } 1 p 1 1 m1 p(m 1 m 1 ) M 1 m 1 p(m 1 m 1 ) { M p 2 m p } p { p 1 { 2 M p + 2 m p 2 M p 2 m p } } 1 p 1 2 m2 p(m 2 m 2 ) M 2 m 2 p(m 2 m 2 ) While, for 0 < p < 1, we have the reverse inequality in Eq (3.33). Proof. Applying Eq (3.24), we have Now, A p B p = A p I + I B p A p I = A p I p λ 1 (A I) p I B p = I p B p λ 2 (I B) p λ 1 (A I) p + λ 2 (I B) p max {λ 1, λ 2 } [A I + I B] p = max {λ 1, λ 2 } (A B) p where, λ 1 and λ 2 are given in Eq (3.31) and Eq (3.32). Similarly, from Eq (3.25), we obtain for (A p B p ) (A B) p max {α 1, α 2 } I where, α 1 and α 2 are given in Eq (3.34) and Eq (3.35). Special cases include from Eq (3.30): { (3.36) A 2 B 2 (M1 + m 1 ) 2 max, (M } 2 + m 2 ) 2 {A B} 2. 4M 1 m 1 4M 2 m 2 (3.37) A 1 B 1 max Similarly, special cases include from Eq (3.33): { (M1 + m 1 ) 2, (M } 2 + m 2 ) 2 {A B} 1. 4M 1 m 1 4M 2 m 2

11 MATRIX EQUALITIES AND INEQUALITIES INVOLVING KHATRI-RAO AND TRACY-SINGH SUMS 11 (3.38) (A 2 B 2 ) (A B) 2 max (3.39) (A 1 B 1 ) (A B) 1 max { 1 4 (M 1 m 1 ) 2, 1 } 4 (M 2 m 2 ) 2 I. { M1 m 1 4M 1 m 1, M2 m 2 4M 2 m 2 Theorem Let A and B be positive definite Hermitian compatibly partitioned matrices. Let m and M be, respectively, the smallest and the largest eigenvalues of A B. Then (a) For p a nonzero integer, we have (3.40) A p B p λ(a B) p, where λ is given by Eq (3.21). While, for 0 < p < 1, we have the reverse inequality in Eq (3.40). (b) For p a nonzero integer, we have (3.41) (A p B p ) (A B) p αi where, α is given by Eq (3.23). While, for 0 < p < 1, we have the reverse inequality in Eq (3.41). Proof. In Eq (3.20) and Eq (3.22), take k = 1 and instead of U, use Z, the selection matrix which satisfy the following property: A B = Z (A B)Z, Z Z = I Then, with Z, A B, Z substituted for U, X, U, we have from Eq (3.20) where, λ is given by Eq (3.21). A p B p = Z (A p B p )Z = Z (A p ΠI + IΠB p )Z Z {A B} p Z Similarly, from Eq (3.22), we obtain Eq (3.41) Special cases include from Eq (3.40): (3.42) A 2 B 2 λ {Z (A B)Z} p = λ(a B) p (M + m)2 4Mm {A B}2 (3.43) A 1 B 1 (M + m)2 4Mm {A B} 1 Similarly, special cases include from Eq (3.41): (3.44) (A 2 B 2 ) (A B) (M m)2 I } I. (3.45) (A 1 B 1 ) (A B) 1 M m Mm {I}

12 12 ZEYAD AL ZHOUR AND ADEM KILICMAN 4. SPECIAL RESULTS ON HADAMARD AND KRONECKER SUMS The results obtained in Section 3 are quite general. Now, we consider some inequalities in a special case which involves non-partitioned matrices A, B and I with the Hadamard product (sum) replacing the Khatri-Rao product (sum) and the Kronecker product (sum) replacing the Tracy-Singh product (sum). As these inequalities can be viewed as a corollary (some of) the proofs are straightforward and alternative to those for the existing inequalities. Definition 4.1. Let A and B be square matrices of order n n.the Hadamard sum is defined as follows: (4.1) A B = A I n + I n B = A I n + B I n = (A + B) I n. Corollary 4.1. Let A > 0. Then (4.2) A A 1 2I. Corollary 4.2. Let A > 0 be a matrix of order n n with eigenvalues contained in the interval between m and M (M m). Then (4.3) A A 1 (m2 + M 2 ) mm {I}. Corollary 4.3. Let A and B be n n positive definite Hermitian matrices and let m and M be, respectively, the smallest and the largest eigenvalues of A B. Then (a) For p a nonzero integer, we have (4.4) A p B p λ(a B) p where, λ is given by Eq (3.21). While, for 0 < p < 1, we have the reverse inequality in Eq (4.4). (b) For p is a nonzero integer, we have (4.5) (A p B p ) (A B) p αi where, α is given by Eq (3.23). While, for 0 < p < 1, we have the reverse inequality in Eq (4.5). Special cases include from Eq (4.4): (4.6) A 2 B 2 (M + m)2 4Mm {A B}2 (4.7) A 1 B 1 (M + m)2 4Mm {A B} 1. Similarly, special cases include from Eq (4.5): (4.8) (A 2 B 2 ) (A B) (M m)2 I M m (4.9) (A 1 B 1 ) (A B) 1 Mm {I}, where results in Eq (4.6), Eq (4.7), and Eq (4.8) are given in [11].

13 MATRIX EQUALITIES AND INEQUALITIES INVOLVING KHATRI-RAO AND TRACY-SINGH SUMS 13 We note that the eigenvalues of A B are the n 2 products of the eigenvalues of A by the eigenvalues of B.Thus if the eigenvalues of A and B are, respectively, ordered by: (4.10) δ 1 δ 2 δ n > 0, η 1 η 2 η n > 0, then in all the previous results in this section M = δ 1 η 1 and m = δ n η n. Thus Eq (4.6) to Eq (4.9) become: (4.11) A 2 B 2 (δ 1η 1 + δ n η n ) 2 4δ 1 η 1 δ n η n {A B} 2 (4.12) A 1 B 1 (δ 1η 1 + δ n η n ) 2 {A B} 1 4δ 1 η 1 δ n η n (4.13) (A 2 B 2 ) (A B) (δ 1η 1 δ n η n ) 2 {I} (4.14) ( A 1 B 1) (A B) 1 ( δ1 η 1 δ n η n ) δ 1 η 1 δ n η n {I}. Corollary 4.4. Let A and B be a n n positive definite Hermitian matrices. Let m 1 and M 1 be, respectively, the smallest and the largest eigenvalues of A I and m 2 and M 2, respectively, the smallest and the largest eigenvalues of I B. Then (a) For p a nonzero integer, we have (4.15) A p B p max {λ 1, λ 2 } (A B) p, where λ 1 and λ 2 are given by Eq (3.31) and Eq (3.32). While, for 0 < p < 1, we have the reverse inequality in Eq (4.15). (b) For p a nonzero integer, we have (4.16) (A p B p ) (A B) p max {α 1, α 2 } I, where α 1 and α 2 are given by Eq (3.34) and Eq (3.35). While, for 0 < p < 1, we have the reverse inequality in Eq (4.16). Note that, the eigenvalues of A I equal the eigenvalues of A and the eigenvalues of I B equal the eigenvalues of B. Corollary 4.5. Let A and B be n n positive definite Hermitian matrices. Let m and M be, respectively, the smallest and the largest eigenvalues of A B. Then (a) For p a nonzero integer, we have (4.17) A p B p λ(a B) p, where, λ is given by Eq (3.21). While, for 0 < p < 1, we have the reverse inequality in Eq (4.17). (b) For p a nonzero integer, we have (4.18) (A p B p ) (A B) p αi, where, α is given by Eq (3.23). While, for 0 < p < 1, we have the reverse inequality in Eq (4.18).

14 14 ZEYAD AL ZHOUR AND ADEM KILICMAN Special cases include from Eq (4.17): (4.19) A 2 B 2 (M + m)2 4Mm {A B}2 (4.20) A 1 B 1 Similarly, special cases include from Eq (4.18): (M + m)2 4Mm {A B} 1 (4.21) (A 2 B 2 ) (A B) (M m)2 I (4.22) (A 1 B 1 ) (A B) 1 M m Mm {I}. We note that the eigenvalues of A B are the n 2 sums of the eigenvalues of A by the eigenvalues of B. Thus if the eigenvalues of A and B are, respectively, ordered by: δ 1 δ 2 δ n > 0, η 1 η 2 η n > 0, then in all previous results of this section M = δ 1 + η 1 and m = δ n + η n. Thus Eq(4.19) to Eq (4.22) become: (4.23) A 2 B 2 (δ 1 + η 1 + δ n + η n ) 2 4(δ 1 + η 1 )(δ n + η n ) {A B}2, (4.24) A 1 B 1 (δ 1 + η 1 + δ n + η n ) 2 4(δ 1 + η 1 )(δ n + η n ) {A B} 1, (4.25) (A 2 B 2 ) (A B) ((δ 1 + η 1 ) (δ n + η n )) 2 I, (4.26) (A 1 B 1 ) (A B) 1 δ1 + η 1 δ n + η n I. (δ 1 + η 1 )(δ n + η n ) Corollary 4.6. Let A 0 and B 0 be compatibly matrices. Then (4.27) (4.28) (i) (A B)(A B) AA BB (ii) (A B) w A w B w, for any positive integer w. Corollary 4.7. Let A and B be matrices of order m m and n n respectively. Then (4.29) (4.30) (a) tr(a B) = n tr(a) + m tr(b) (b) A B p p n A p + p m B p, (4.31) (c) where A p = [tr A p ] 1/p, 1 p <. e A B = e A e B

15 MATRIX EQUALITIES AND INEQUALITIES INVOLVING KHATRI-RAO AND TRACY-SINGH SUMS 15 Corollary 4.8. Let A and B be non singular matrices of order m m and n n, respectively. Then (4.32) (4.33) (4.34) (i) (A B) 1 = (A 1 B 1 ) 1 (A 1 B 1 ) (ii) (A B) 1 = (A 1 I n )(A 1 B 1 ) 1 (I m B 1 ) (iii) (A B) 1 = (I m B 1 )(A 1 B 1 ) 1 (A 1 I n ) In [1], Ando proved the following inequality; (4.35) A B (A p I) 1 p (B q I) 1 q, where A and B are positive definite matrices and p, q 1with 1/p + 1/q = 1. If is a unitarily invariant norm and is the spectral norm, Horn and Johnson in [3] proved the following three conditions are equivalent: (4.36) (i) (ii) (iii) A A AB A B A B A B for all matrices A and B. In [2], Hiai and Zhan proved the following inequalities: (4.37) AB A B A + B A + B and A B A B A + B A + B for any invariant norm with diag(1, 0,..., 0) 1 and A, B are nonzero positive definite matrices. We have an application to generalize the inequalities in Eq (4.37) involving the Hadamard product (sum) and the Kronecker product (sum). Theorem 4.9. Let be a unitarily invariant norm with diag(1, 0,..., 0) 1 and A and B be nonzero positive definite matrices. Then (4.38) A B A B A B A + B. Proof. Let be the spectral norm and applying Eq (4.35) to A/ A I, B/ B I and using the Young inequality for scalars, we get ( ) [( ) p 1 A B A I] [( p A B A ( ) p A I + 1 B B ( B ) q I] 1 q ) q I We choose 1 p = A [ A + B ] 1 p A q B 1 ( ) A I + 1 ( ) B I p A q B { ( ) 1 A = + 1 ( )} B I p A q B and 1 q = B [ A + B ].

16 16 ZEYAD AL ZHOUR AND ADEM KILICMAN Since A A and B B thanks to diag(1, 0,..., 0) 1, we obtain { } A B (4.39) A B (A + B) I A + B { } A B (A B) A + B Hence, A B A B A + B A B or A B A B A B A + B Corollary Let be a unitarily invariant norm with diag(1, 0,..., 0) 1 and A and B be nonzero positive definite matrices. Then (4.40) A B A B Proof. Applying Eq (2.7) and Eq (4.39), we have K (A B)K A B A + B. A B A + B K (A B)K and K A B (A B)K A + B K (A B)K. Provided that is unitarily invariant norm, we get the result. REFERENCES [1] T. ANDO, Concavity of certain maps on positive definite matrices, and applications to Hadamard products, Lin. Alg. and its Appl., 26 (1979), [2] F. HIAI AND X. ZHAN, Submultiplicativity vs subadditivity for unitarily invariant norms, Lin. Alg. and its Appl., 377 (2004), [3] R.A. HORN AND C.R. JOHNSON, Hadamard and conventional submultiplicativity for unitarily invariant norms on matrices, Lin. Multilinear Alg., 20 (1987), [4] C.G. KHATRI AND C.R. RAO, Solutions to some functional equations and their applications to characterization of probability distributions, Sankhya, 30 (1968), [5] R.H. KONING, H. NEUDECKER AND T. WANSBEEK, Block Kronecker products and the vecb operator, Lin. Alg. and its Appl., 149 (1991), [6] S. LIU, Contributions to matrix calculus and applications in econometrics, Tinbergen Institute Research Series, 106, Thesis publishers, Amsterdam, The North land, (1995). [7] S. LIU, Matrix results on the Khatri-Rao and Tracy-Singh products, Lin. Alg. and its Appl., 289 (1999), [8] S. LIU, Several inequalities involving Khatri-Rao products of positive semi definite matrices, Lin. Alg. and its Appl., 354 (2002), [9] S. LIU, Inequalities involving Hadamard products of positive semi definite matrices, 243 (2002), [10] B. MOND AND J.E. PEČARIĆ, Matrix inequalities for convex functions, J. of Math. Anal. and Appl., 209 (1997),

17 MATRIX EQUALITIES AND INEQUALITIES INVOLVING KHATRI-RAO AND TRACY-SINGH SUMS 17 [11] B. MOND AND J.E. PEČARIĆ, Inequalities for the Hadamard product of matrices, SIAM J. Matrix Anal. and Appl., 19 (1998), [12] D.S. TRACY AND R.P. SINGH, A new matrix product and its applications in matrix differentiation, Statist. Neerlandica, 26 (1972), [13] C.R. RAO AND J. KLEFFE, Estimation of Variance Components and Applications, North-Holland, Amsterdam, The Netherlands, (1988). [14] C.R. RAO AND M.B. RAO, Matrix Algebra and its Applications to Statistics and Econometrics, World Scientific, Singapore, (1998). [15] G. VISICK, A quantitative version of the observation that the Hadamard product is a principle submatrix of the Kronecker product, Lin. Alg. and its Appl., 304 (2000), [16] SONG-GUI WANG AND WAI-CHEUNG IP, A matrix version of the Wielandt inequality and its applications to statistics, Lin. Alg. and its Appl., 296 (1999),