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Several matrix trace inequalities on Hermitian and skew-Hermitian matrices
Journal of Inequalities and Applications volume 2014, Article number: 358 (2014)
In this paper, we present several matrix trace inequalities on Hermitian and skew-Hermitian matrices, which play an important role in designing and analyzing interior-point methods (IPMs) for semidefinite optimization (SDO).
SDO is the generalization of linear optimization (LO), which is convex optimization over the intersection of an affine set and the cone of positive semidefinite matrices. Several IPMs designed for LO have been successfully extended to SDO [1, 2]. Some matrix trace inequalities are developed and applied in the analysis of IPMs for SDO (see [3–12]). In , Yang proved the arithmetic mean-geometric mean inequality for positive definite matrices, which was an open question proposed by Bellman in ; Neudecke used a different method in  to show a slightly relaxed version of Yang’s result in ; In , Coope considered alternative proofs of some simple matrix trace inequalities in [3–5] and further studied properties of products of Hermitian and positive (semi)definite matrices; In , Yang gave a new proof of the result obtained by Yang in  and extended it to a generalized positive semidefinite matrix. Based on the work in [3–5], Chang established a matrix trace inequality for products of Hermitian matrices in , which partly answers a conjecture proposed by Bellman in . In addition, Yang gave a matrix trace inequality for products of positive semidefinite matrices in ; In , Yang et al. established a matrix trace inequality for positive semidefinite matrices, which improved the result given by Yang in . Although there have been many results on matrix trace inequality, some important matrix trace inequality problems have not been fully solved. In this paper, we will provide several matrix trace inequalities on Hermitian and skew-Hermitian matrices, which play an important role in designing and analyzing IPMs for SDO.
This paper is organized as follows: In Section 2, a matrix trace inequality on Hermitian and skew-Hermitian matrices is provided, and its simple proof is given by using an elementary method. However, it is difficult for us to use this method to deal with the high-dimensional case. Based on Lemmas 2 and 3 in Section 2, the high-dimensional case will be shown as Corollary 2 in Section 3. In Section 4, some conclusions are made.
The following notations are used throughout the paper. N, , C, and denote the set of natural numbers, the set of nonnegative real numbers, the set of complex numbers and the set of vectors with n components, respectively. is the space of all matrices over C. Define
where . The vector inner product of and is defined by . and denote modules for complex numbers and the 2-norm for vectors, respectively. For , represents the conjugate transpose of A, () means that is positive semidefinite (positive definite). For any Hermitian positive definite matrix Q, the expression denotes the Hermitian square root of Q. Similarly, the power can be defined for any and .
2 Preliminary results
In this section, we will present a matrix trace inequality on Hermitian and skew-Hermitian matrices.
Lemma 1 Let and M be Hermitian and skew-Hermitian matrices, respectively. Then
Proof From , there exists a unitary matrix U such that
where is a skew-symmetric matrix.
Without loss of generality, let
where . Then
it follows that
Suppose and are two roots of (1), where . According to Weda’s theorem, we obtain
From (2), it is easy to compute that
On the other hand, it is clear that
According to (4), it follows that
Thus, (5) can be written as
This, together with (3), yields
Remark 1 In this lemma, Weda’s theorem is applied to prove the matrix trace inequality of Hermitian and skew-Hermitian matrices. However, it is difficult for us to extend this result to the high-dimensional case, because it is too complex for us to compute the matrix traces of Hermitian and skew-Hermitian matrices by using Weda’s theorem, when .
In order to prove the high-dimensional case of Lemma 1, we need to give the following two lemmas.
Lemma 2 
Let . Then , , .
Lemma 3 
Let with eigenvalues and their corresponding orthonormal eigenvectors , respectively. Define
for any with .
3 Main results
In this section, we develop several matrix trace inequalities on Hermitian and skew-Hermitian matrices. Furthermore, Lemma 1 is extended to the high-dimensional case as Corollary 2.
Theorem 1 Let
If and are singular values of A and eigenvalues of H, respectively. They are arranged in such a way that
Proof From the definition of singular values of A, it follows that
are eigenvalues of , and their corresponding orthonormal eigenvectors of are denoted as , respectively. Suppose are orthonormal eigenvectors of H, and
are eigenvalues of H corresponding to these orthonormal eigenvectors, respectively. Define
where . It is clear that
According to the dimensional formula, it follows that
This implies that
From Lemma 3, it follows that
for any with . According to Lemma 2, we obtain
where . This completes the proof. □
Remark 2 In this section, we prove Theorem 1 by using a simple and elementary method, which is slightly different from the proof method in . This theorem reveals the relationship between singular values and eigenvalue of matrices. Based on it, several matrix trace inequalities on Hermitian and skew-Hermitian matrices will be obtained immediately.
Theorem 2 Let , . The following matrix trace inequalities are satisfied.
If , then
If , then
Proof (I) From Theorem 1, it follows that
Similar to the proof of (I), we can easily verify that (II) and (III) hold. This completes the proof. □
where and be Hermitian and skew-Hermitian matrices, respectively. By using Theorem 2, the following conclusions will be obtained immediately.
Corollary 1 Let and be Hermitian and skew-Hermitian matrices, respectively. The following matrix trace inequalities are satisfied.
If , then
If , then
Based on Corollary 1, the following results can be obtained when . This implies that the proof of the general case of Lemma 1 is completed.
Corollary 2 Let and be a Hermitian and skew-Hermitian matrices, respectively. The following matrix trace inequalities are satisfied.
If , then
If , then
This paper proves several matrix trace inequalities on Hermitian and skew-Hermitian matrices. These matrix trace inequalities can be applied to design and analyze interior-point methods (IPMs) for semidefinite optimization (SDO). In addition, matrix trace inequalities have many potential applications in control theory, for example, stabilization of time-delay systems (see [15–18]).
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The authors thank the editor and the referees for their valuable suggestions to improve the quality of this paper. This work was partially supported by the National Natural Science Foundation of China (No. 11371006), Natural Science Foundation of Heilongjiang Province (No. A201416), the Undergraduate Innovation and Entrepreneurship Program of Heilongjiang University (No. 2014SX04) and the Laboratory Project of Heilongjiang University.
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
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Gao, X., Wang, G., Zhang, X. et al. Several matrix trace inequalities on Hermitian and skew-Hermitian matrices. J Inequal Appl 2014, 358 (2014). https://doi.org/10.1186/1029-242X-2014-358
- Hermitian matrix
- skew-Hermitian matrix
- Hermitian positive definite matrix
- semidefinite optimization
- interior-point methods