The further generalization on the inequalities for Hadamard products of any number of invertible Hermitian matrices
© Chen et al.; licensee Springer. 2014
Received: 15 May 2014
Accepted: 18 November 2014
Published: 2 December 2014
Without ‘positive definiteness’ demanded in the present papers, the forward and reverse inequalities for Hadamard products of any number of invertible Hermitian matrices are obtained, and the sufficient and necessary conditions for the equations in these inequalities are given. As Hermitian positive matrices naturally satisfy the added constraints, these results generalize and improve the corresponding results in the present papers. Beyond that, with no demand of ‘positive definiteness’, these forward and backward inequalities are not determined mutually any longer.
KeywordsHermitian matrix Hadamard product matrix inequality equation condition
Throughout the paper, we assume is the set of complex matrices, I is a identity matrix, is a diagonal matrix with 1 at its th position and 0 elsewhere, and is a selection matrix. stands for the conjugate transpose of . The matrix A is Hermitian if , denoted by . Furthermore, we denote by and the sets of Hermitian semi-positive matrices and Hermitian positive matrices, respectively. Recall that A, B are said to have the inequality or , if . In particular, (resp. ), denoted by (resp. ), and denoted by the square root of A. For a positive integer k, we have , . Especially, . For , denotes the submatrix of A lying in rows indexed by α and the columns indexed by β and . The Hadamard and Kronecker products of , are defined as and , respectively. If , , , and , then X is said to be a Moore-Penrose generalized inverse of A, denoted by .
Recall that if every diagonal element of is 1, then R is said to be a correlation matrix, in symbol (see ). The invertible matrix implies , thus the set of invertible matrices .
Meanwhile, Styan pointed out ‘A matrix-theoretic proof of Theorem 4.1 would be of interest.’ (see ).
In 1979, Ando  obtained the following.
Proposition 1.1 (see [, Theorem 20])
When , the inequality (1.5) implies (1.3). Ando also made clear, by applying the method of Proposition 1.1, that one has the following.
Proposition 1.2 (see [, p.239])
In 2000, Zhang showed the following.
Proposition 1.3 (see [, Application 4])
As applications, (1.2) and (1.4) were also given by Visick in 2000 (see [, Theorem 20]).
For , by , one has . Because of the commutativity of the Hadamard product, the inequality (1.9) is equivalent to [, Proposition 1], [, Corollary 1(5)], it could be viewed as a generalization of (1.4) over Hermitian semi-positive matrices.
For , (1.6) and (1.7) are the forward and backward inequalities to each other, and determined mutually as well (Theorem 2.6),  has ever called them as companion inequalities. Of course, (1.1) and (1.2), (1.3) and (1.4) are also the companion inequalities determined by each other (Lemma 2.4). In fact, [, Theorem 1] shows us the backward inequality which is companied to (1.9).
Papers [1–10] all discuss on Hermitian (semi-)positive matrices. However, the following Example 1.4 illustrates that the condition ‘positive definiteness’ is not necessary for the inequality (1.7) to hold.
Example 1.5 indicates that, in general, the matrix inequality (1.7) does not hold for all invertible Hermitian matrices. Hence we will add some constraint conditions in our discussion.
In this paper, without the ‘positive definiteness’ demanded in the inequality (1.5), we will show inequalities and their companion forms for any number of invertible Hermitian matrices and get sufficient and necessary conditions for the equations in the related inequalities to hold. In view of [12, 13]etc., we see the discussion of the equation conditions for the inequalities is significant. Further, Hermitian positive matrices satisfy the added constraints naturally, thus these results generalize and improve the corresponding results in the present literature. However, with no demand of ‘positive definiteness’, the new forward and backward companion inequalities are not determined mutually.
which indicates (2.7) and (2.8) hold for .
which shows the determined by satisfies (2.8), then (2.7) holds.
thus the determined by satisfies (2.8), then (2.7) follows by (2.11). □
The proof method of Lemma 2.1 plays a great role in discussing the matrix inequalities for Khatri-Rao products of any finite number of positive matrices (see [, Lemma 2.1], [, Lemmas 2.1 and 2.2]). Recently, [, Theorem 3] has also discussed a similar problem to Lemma 2.1, but our results (2.7) and (2.8) should be more convenient in applications.
Lemma 2.2 Let and be both invertible. Then both of and are invertible as well, and .
by comparing with (2.12), it follows that is invertible and . □
When , it is natural that is invertible, hence we could obtain [, formula (4)] again by Lemma 2.2.
Moreover, the equation in (2.13) holds if and only if .
For a Hermitian positive matrix A, one has ; then we get [, Theorem 1(7)] again by Lemma 2.3.
Lemmas 2.2-2.4 will play an important role in the discussion.
Moreover, the equation in (1.5) holds if and only if the one in (2.17) holds.
Proof As , by (2.6), . Taking , in view of Proposition 1.1, yields the inequality (1.5), that is, , then by Lemma 2.4, , which shows (2.17) holds, meanwhile, the equation in (1.5) holds; therefore the one in (2.17) holds. □
Theorem 2.5 not only leads to the backward inequality (2.17) of (1.5) (in this case, the inequalities (1.5) and (2.17) are mutually determined), but it also shows us that the inequalities (1.1) and (1.2), (1.3) and (1.4) given by Styan are companied and determined by each other (the case of in Theorem 2.5).
By applying Lemma 2.4, Propositions 1.2 and 1.3, with a similar discussion as Theorem 2.5, we have the following.
Theorem 2.6 Let . Then the inequalities (1.6) and (1.7) are companied and determined by each other, and the equation in (1.6) holds; therefore the one in (1.7) holds.
3 Main results
so (3.1) holds.
From Theorems 2.5 and 2.6, we see the inequalities (1.1) and (1.2), (1.3) and (1.4), and the general ones (1.5) and (2.17), (1.6) and (1.7) are companied and determined by each other, hence one of the companion inequalities could be obtained from the other one immediately. However, the following example indicates that the matrix inequality (3.1) is no longer equivalent to its backward inequality, without ‘positive definiteness’.
thus the inequality (3.4) follows by (3.3).
From Lemma 2.3 and the proof course as above, with a similar discussion as Theorem 3.1, we see the condition for the equation in (3.4) is determined by (3.5). □
From above, the case discussed here is without ‘positive definiteness’, which is different from [1–10], and in form, the inequalities (3.1) and (3.4) are the reverses to each other; however, Theorems 3.1 and 3.3, and Example 3.2 indicate that their constraints are different, so (3.1) and (3.4) are not determined by each other any longer.
When , we could obtain [, Corollaries 2 and 3] from Theorems 3.1 and 3.3 immediately.
- (i)if C is invertible and , then is also invertible and the inequality (1.5) holds. Meanwhile, we have the equation in (1.5) if and only if(3.6)
- (ii)if , then is invertible and the inequality (2.17) holds. Meanwhile, the equation in (2.17) holds if and only if(3.7)
then we could obtain the results by taking () in Theorems 3.1 and 3.3. □
Corollary 3.4 indicates that, without ‘positive definiteness’, not only the inequality (1.5) still holds under some constraints, but also its reverse inequality (2.17) still holds as well. Clearly their constraints are different.
- (i)if , then is invertible and(3.8)
- (ii)if C is invertible and , then is also invertible and(3.10)
Proof By the assumption, () is invertible with all diagonal elements 1, so , , then in view of (3.1), (3.2), (3.4), and (3.5) we have the conclusions. □
For , by (2.6), they satisfy the constraints demanded in Theorems 3.1 and 3.3 naturally. Hence similar to Theorem 2.6, by Lemma 2.4, we have the following.
Theorem 3.6 Let () and be the one as in (2.7) and (2.8), then both of inequalities (3.1) and (3.4) hold, and the equation in (3.1) holds iff the one in (3.4) holds iff (3.2) holds iff (3.5) holds.
Now in view of Theorem 2.5 and (2.17), we see the inequality (3.12) obtained from (1.8) is different from the one in (1.5). When (), by Theorem 3.6, we have the following.
Corollary 3.7 Let and be the one as in (2.7) and (2.8). Then both of inequalities (1.5) and (2.17) hold, and the equation in (1.5) holds iff the one in (2.17) holds iff (3.6) holds iff (3.7) holds.
By applying Theorem 3.6 and Corollaries 3.4, 3.5, we are led to the following conclusion.
Corollary 3.8 Let (), be the one as in (2.7) and (2.8). Then both of inequalities (3.8) and (3.10) hold, moreover, the equation in (3.8) holds if and only if the one in (3.10) holds; thus (3.9) and (3.11) hold.
When , the companion inequalities (1.1)-(1.4), (1.6), and (1.7), and their equation conditions are obtained.
The work is supported by the National Natural Science Foundation of China (No. 61373140), the Natural Science Foundation of Fujian Province (No. 2013J00102), the middle-aged research item in Education Committee of Fujian Province (No. JA14277), the key item of Hercynian building for the colleges and universities service in Fujian Province (2008HX03) and the teaching reformation project of Putian University (JG201415).
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