Inequalities Involving Khatri-Rao Products of Positive Semi-definite Matrices

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1 Applied Mathematics E-Notes, (00), c ISSN Available free at mirror sites of amen/ Inequalities Involving Khatri-Rao Products of Positive Semi-definite Matrices Xian Zhang, Zhong-peng Yang, Chong-guang Cao Received 0 August 001 Abstract Several inequalities involving the Khatri-Rao products of two four-block positive definite real symmetric matrices are established by Liu in [1]. We extend these to two general positive semi-definite real symmetric block matrices necessary sufficient conditions under which these inequalities become equalities are presented. Let S(m) be the set of all real symmetric matrices of order m. Consider matrices M S(m) N S(p) which are partitioned as follows M = M M 1t M 1t... M tt,n= N N 1t N 1t... N tt where M ii S(m i )N ii S(p i )fori =1,,,t.Obviously,, (1) t m i = m, i=1 t p i = p. i=1 We denote by M N =(M ij N ij ) ij M N =(M ij N) ij = (M ij N kl ) kl ij the Khatri-Rao Tracy-Singh products of M N respectively, where represents the Kronecker product. Obviously, t M N S(mp), M N S m i p i. Mathematics Subject Classifications: 15A45, 15A69. Department of Mathematics, Heilongjiang University, Harbin, Heilongjiang, , P. R. China. Department of Mathematics, Putian College, Putian, Fujian, , P. R. China. Department of Mathematics, Heilongjiang University, Harbin, Heilongjiang , P. R. China i=1 117

2 118 Matrix Inequalities When M N are positive definite real symmetric matrices t =, the following inequalities are obtained by Liu in [1]: (M N) 1 M 1 N 1 ; () M 1 N 1 (λ 1 + λ mp ) 4λ 1 λ mp (M N) 1 ; (3) M N (M 1 N 1 ) 1 ( λ 1 λ mp ) I; (4) M N (λ 1 + λ mp ) 4λ 1 λ mp (M N) ; (5) M N (M N) 1 4 (λ 1 λ mp ) I; (6) (M N ) 1/ λ 1 + λ mp λ 1 λ mp (M N); (7) (M N ) 1/ M N (λ 1 λ mp ) I, (8) 4(λ 1 + λ mp ) where λ 1 λ mp are the largest smallest eigenvalue of M N respectively, A B (or B A) meansthata B is positive semi-definite. We remark that the inequality (6) is erroneously printed as (M N) M N 1 4 (λ 1 λ mp ) I in [1, Theorem 8]). We remark further that conditions for equalities in ()-(8) are not known. The purpose of this paper is to extend these inequalities for general block matrices. We also find necessary sufficient conditions for equalities to hold. Liu [1, p.69] also shows that the Khatri-Rao product can be viewed as a generalization of the Hadamard product. Therefore, our results can also be viewed as a generalization of those corresponding inequalities involving the Hadamard product, see e.g., [3, (1.4), (1.5), (.14), (.15), (.19), (.0)]. For a matrix A S(m), we denote by λ(a) τ(a) the largest smallest nonzero eigenvalue of A respectively. Let R(A) be the column space of matrix A. We denote the n n identity matrix by I n,orbyi when the order of matrix is clear. Let S + (m) S 0 + (m) be the set of all positive definite semi-definite real symmetric matrices of order m respectively. LEMMA 1. ([1, Theorem 1 (a)(b)]) If A B are compatibly partitioned, then (A B)(C D) =(AC) (BD) (9) (A B) + = A + B +, (10) where A + is the Moore-Penrose inverse of A. LEMMA. Let A B be compatibly partitioned matrices, then (A B) = A B.

3 Zhang et al. 119 Indeed, (A B) = = = (A ij B) ij = (Aij B kl ) kl (Aij B kl ) kl A ij B kl = A B. lk ij ji = (A ij B kl ) lk ji = A ij B ji LEMMA 3. Suppose A S(m) B S(p). Then i) A B S(mp), λ(a B) =λ(a)λ(b), τ(a B) =τ(a)τ(b), (A B) n = A n B n foranypositiveintegernumbern; ii) A B S 0 + (mp) ifa S+ 0 (m) B S+ 0 (p); iii) A B S + (mp) ifa S + (m) B S + (p). PROOF. Let A = UA D AU A B = UB D BU B be the spectral decompositions of A B respectively. Then using (9) Lemma, A B = (UAD A U A ) (UBD B U B ) = (UA UB)(D A D B )(U A U B ) = (U A U B ) (D A D B )(U A U B ) (11) (U A U B ) (U A U B ) = (UA UB)(U A U B ) = (UAU A ) (UBU B ). (1) Substituting U A U A = I m U B U B = I p into (1), we see that (U A U B ) (U A U B )=I mp. (13) Combining (13) (11) completes the proof. THEOREM 1. There exists a real matrix Z of order mp t m i p i such that Z Z = I i=1 A B = Z (A B)Z (14) for any A S(m) B S(p) which are partitioned as in (1). PROOF. Let Z i = O i1... Q i,i 1 I mi p i O i,i+1... Q it,i=1,,..., t, (15) where O ik isthezeromatrixoforderm i p i m i p k for any k = i. Then Z i Z i = I Z i(a ij B)Z i = Z i(a ij B kl ) kl Z j = A ij B ij,i,j=1,,..., t. ji

4 10 Matrix Inequalities Letting Z 1 Z =..., (16) the result then follows by a direct computation. THEOREM. Suppose Z is defined as in Theorem 1, A S 0 + (mp), W = λ(a), w = τ(a), R(Z) R(A). (17) Then the following conclusions hold. (i) (Z AZ) + Z A + Z, the equality holds if, only if, R(Z) =R(AZ). (ii) Z A + (W +w) Z 4Ww (Z AZ) +, the equality holds if, only if, Z AZ = W +w I Z A + Z = W +w Ww I. (iii) Z AZ (Z A + Z) + ( W w) I, the equality holds if, only if, W = w or Z AZ =(W + w Ww)I, Z A + Z = 1 I. (18) Ww (W +w) (iv) Z A Z 4Ww (Z AZ), the equality holds if, only if, Z AZ = Ww W +w I Z A Z = WwI. (v) Z A Z (Z AZ) 1 4 (W w) I, the equality holds if, only if, W = w or Z AZ = W + w I, Z A Z = W + w I. (19) or (vi) (Z A Z) 1/ Z AZ Z A Z = Z t (W w) 4(W +w) I, the equality holds if, only if, W = w (W + w) 4 I, Z AZ = W + w +6Ww I. (0) 4(W + w) PROOF. It follows from (15) (16) that Z + = Z ZZ I. This, together with [, (3), Propositions 3.1, 3., ], yields the conclusions (i) (v). Now we prove (vi). Combining [, (4)] Z Z = I yields = (Z A Z) 1/ Z AZ (Z AZ) 1/ 1 W + w Z A Z Ww W + w I (W w) 4(W + w) I 1 W + w (Z A Z) 1/ I W + w (W w) 4(W + w) I, the above inequalities become equalities if only if W = w or (0). THEOREM 3. Suppose M S 0 + (m) N S+ 0 (p) are partitioned as in (1), Z is definedasintheorem1,w = λ(m)λ(n), w = τ(m)τ(n), R(Z) R(M N). (1)

5 Zhang et al. 11 Then the following conclusions hold. (i) (M N) + M + N +, the equality holds if, only if, R(Z) =R((M N)Z). (ii) M + N + (W +w) 4Ww (M N)+, the equality holds if, only if, M N = W +w I M + N + = W +w Ww I. (iii) M N (M + N + ) + ( W w) I, the equality holds if, only if, W = w or M N =(W + w Ww)I, M + N + = 1 I. () Ww (iv) M N (W +w) 4Ww, the equality holds if, only if, M N = Ww W +w I M N = WwI. (v) M N (M N) 1 4 (W w) I, the equality holds if, only if, W = w or M N = W + w I, M N = W + w I. (3) or (vi) (M N ) 1/ M N M N = (W + w) 4 (W w) 4(W +w) I, the equality holds if, only if, W = w I, M N = W + w +6Ww I. (4) 4(W + w) W +w (vii) (M N ) 1/ W +w (M N), the equality holds if, only if, M N = Ww I M N = WwI. PROOF. Noting i) ii) of Lemma 3 (10) substituting A by M N in Theorem, we can obtain the conclusions (i) (vi). Furthermore, the conclusion (vii) follows from (iv) [, Proposition.3]. REMARK 1. If t =,M S + (m) N S + (p), then M N automatically satisfy the assumptions of Theorem 3 by applying iii) of Lemma 3, hence Theorem 3 extends the inequalities () (8). REMARK. We have shown that the condition (1) is sufficient for the inequalities stated in Theorem 3. The following two examples will show that this is not a necessary condition. It is still an open problem to determine a sufficient necessary condition under which these inequalities stated in Theorem 3 hold. EXAMPLE 1. Consider matrices M = ,N= (5) It is easy to verify that M,N S 0 + (3). According (15) (16), we can obtain Z = (6)

6 1 Matrix Inequalities Furthermore, we can easily show that matrices M, N Z do not satisfy the condition (1) the inequalities stated in Theorem 3. Indeed, W =1.704 w = Furthermore, M + = ,N + = , M + N + = , (M N) + = , M = ,N = , M N = , (M N ) 1/ = , (M + N + ) + = , M + N + (M N) + = O, (W + w) 4Ww (M N)+ M + N + = O, ( W w) I M N (M + N + ) + = O, (W + w) 4Ww (M N) M N = O, (W w) 4(W + w) I (M N ) 1/ M N = O, (W w) I M N (M N) = O,

7 Zhang et al. 13 W + w Ww (M N) (M N ) 1/ = O. EXAMPLE. Consider matrices M = ,N= (7) It is easy to verify that M,N S 0 + (3). According to (15) (16), we can see that the matrix Z possesses the form (6). Furthermore, we can easily show that matrices M, N Z do not satisfy the condition (1), but they satisfy the inequalities stated in Theorem 3. Indeed, W = w = Furthermore, M + = 0 0 0, N + = Then M + N + = , (M N) + = , M = 0 0 0,N = , M N = , (M N ) 1/ = , (M + N + ) + = , M + N + (M N) + = O, (W + w) 4Ww (M N)+ M + N + = O, ( W w) I M N (M + N + ) + = >O,

8 14 Matrix Inequalities (W + w) 4Ww (M N) M N =10 3 (W w) 4(W + w) I (M N ) 1/ M N = 1 4 (W w) I M N (M N) = O, W + w Ww (M N ) 1/ = O. >O, >O, REMARK 3. Since Theorem 3 can be obtained by substituting A with M N in Theorem, the condition (17) is not necessary for the inequalities stated in Theorem toholdbychoosinga = M N, wherem N are definedasinexamples1 respectively. It is also an open problem to determine a sufficient necessary condition under which these inequalities stated in Theorem hold. ACKNOWLEDGMENT. We wish to thank the referee for his helpful comments. The first third authors are partially supported by the NSF of Heilongjiang (No. A01-07) the NSF of Heilongjiang Education Committee (No ). The second author is partially supported by the NSF of Fujian Education Committee. References [1] S. Z. Liu, Matrix results on the Khatri-Rao Tracy-Singh products, Lin. Alg. Appl., 89(1999), [] S. Liu W. Polasek, Equality conditions for matrix Kantorovich-Type inequalities, J. Math. Anal. Appl., 1(1997), [3] B. Mond J. E. Pecaric, Inequalities for the Hadamard product of matrix, SIAM. J. Matrix Anal. Appl., 19(1)(1998),