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Several matrix trace inequalities on Hermitian and skewHermitian matrices
Journal of Inequalities and Applications volume 2014, Article number: 358 (2014)
Abstract
In this paper, we present several matrix trace inequalities on Hermitian and skewHermitian matrices, which play an important role in designing and analyzing interiorpoint methods (IPMs) for semidefinite optimization (SDO).
1 Introduction
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 [3], Yang proved the arithmetic meangeometric mean inequality for positive definite matrices, which was an open question proposed by Bellman in [4]; Neudecke used a different method in [5] to show a slightly relaxed version of Yang’s result in [3]; In [6], 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 [7], Yang gave a new proof of the result obtained by Yang in [3] 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 [8], which partly answers a conjecture proposed by Bellman in [4]. In addition, Yang gave a matrix trace inequality for products of positive semidefinite matrices in [9]; In [10], Yang et al. established a matrix trace inequality for positive semidefinite matrices, which improved the result given by Yang in [9]. 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 skewHermitian 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 2\times 2 Hermitian and skewHermitian 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 highdimensional case. Based on Lemmas 2 and 3 in Section 2, the highdimensional 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, {\mathbf{R}}^{+}, C, and {\mathbf{C}}^{n} 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. {\mathbf{C}}^{n\times n} is the space of all n\times n matrices over C. Define
where {v}_{1},{v}_{2},\dots ,{v}_{n}\in {\mathbf{C}}^{n}. The vector inner product of \alpha \in {\mathbf{C}}^{n} and \beta \in {\mathbf{C}}^{n} is defined by (\alpha ,\beta ). \cdot  and \parallel \cdot \parallel denote modules for complex numbers and the 2norm for vectors, respectively. For A,B\in {\mathbf{C}}^{n\times n}, {A}^{\ast} represents the conjugate transpose of A, A\ge B (A>B) means that AB is positive semidefinite (positive definite). For any Hermitian positive definite matrix Q, the expression {Q}^{\frac{1}{2}} denotes the Hermitian square root of Q. Similarly, the power {Q}^{r} can be defined for any Q>0 and r\in \mathbf{R}.
2 Preliminary results
In this section, we will present a matrix trace inequality on 2\times 2 Hermitian and skewHermitian matrices.
Lemma 1 Let N>0 and M be 2\times 2 Hermitian and skewHermitian matrices, respectively. Then
Proof From N>0, there exists a unitary matrix U such that
where
Hence,
and
where {U}^{\ast}MU is a skewsymmetric matrix.
Without loss of generality, let
where {n}_{1}\ge {n}_{2}>0. Then
Thus
From
it follows that
Suppose {\lambda}_{1} and {\lambda}_{2} are two roots of (1), where {\lambda}_{1}\ge {\lambda}_{2}. 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
This implies
Thus, (5) can be written as
This, together with (3), yields
i.e.,
Hence,
□
Remark 1 In this lemma, Weda’s theorem is applied to prove the matrix trace inequality of 2\times 2 Hermitian and skewHermitian matrices. However, it is difficult for us to extend this result to the highdimensional case, because it is too complex for us to compute the matrix traces of n\times n Hermitian and skewHermitian matrices by using Weda’s theorem, when n\ge 3.
In order to prove the highdimensional case of Lemma 1, we need to give the following two lemmas.
Lemma 2 [13]
Let A\in {\mathbf{C}}^{n\times n}. Then {x}^{\ast}(A+{A}^{\ast})x\le 2\sqrt{{x}^{\ast}{A}^{\ast}Ax}, \mathrm{\forall}x\in {\mathbf{C}}^{n}, \parallel x\parallel =1.
Lemma 3 [13]
Let H={H}^{\ast}\in {\mathbf{C}}^{n\times n} with eigenvalues {\lambda}_{1}(H)\ge {\lambda}_{2}(H)\ge \cdots \ge {\lambda}_{n}(H) and their corresponding orthonormal eigenvectors {v}_{1},{v}_{2},\dots ,{v}_{n}, respectively. Define
Then
for any v\in V with \parallel v\parallel =1.
3 Main results
In this section, we develop several matrix trace inequalities on Hermitian and skewHermitian matrices. Furthermore, Lemma 1 is extended to the highdimensional case as Corollary 2.
Theorem 1 Let
If {\sigma}_{1}(A),{\sigma}_{2}(A),\dots ,{\sigma}_{n}(A) and {\lambda}_{1}(H),{\lambda}_{2}(H),\dots ,{\lambda}_{n}(H) are singular values of A and eigenvalues of H, respectively. They are arranged in such a way that
Then
where k=1,2,\dots ,n.
Proof From the definition of singular values of A, it follows that
are eigenvalues of {A}^{\ast}A, and their corresponding orthonormal eigenvectors of {A}^{\ast}A are denoted as {u}_{1},{u}_{2},\dots ,{u}_{n}, respectively. Suppose {v}_{1},{v}_{2},\dots ,{v}_{n} are orthonormal eigenvectors of H, and
are eigenvalues of H corresponding to these orthonormal eigenvectors, respectively. Define
and
where k=1,2,\dots ,n. It is clear that
Then
According to the dimensional formula, it follows that
This implies that
From Lemma 3, it follows that
for any w\in {W}_{1k}\cap {W}_{kn} with \parallel w\parallel =1. According to Lemma 2, we obtain
where k=1,2,\dots ,n. 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 [14]. This theorem reveals the relationship between singular values and eigenvalue of matrices. Based on it, several matrix trace inequalities on Hermitian and skewHermitian matrices will be obtained immediately.
Theorem 2 Let A\in {\mathbf{C}}^{n\times n}, H=\frac{A+{A}^{\ast}}{2}. The following matrix trace inequalities are satisfied.

(I)
If H\ge 0, then
tr\left({\left({A}^{\ast}A\right)}^{r}\right)\ge tr\left({H}^{2r}\right),\phantom{\rule{1em}{0ex}}\mathrm{\forall}r\in {\mathbf{R}}^{+}. 
(II)
If H>0, then
tr\left({\left({A}^{\ast}A\right)}^{r}\right)\le tr\left({H}^{2r}\right),\phantom{\rule{1em}{0ex}}\mathrm{\forall}r\in {\mathbf{R}}^{+}. 
(III)
tr\left({\left({A}^{\ast}A\right)}^{\frac{2k+1}{2}}\right)\ge tr\left({H}^{2k+1}\right),\phantom{\rule{1em}{0ex}}\mathrm{\forall}k\in \mathbf{N}.
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. □
Let
where N\in {\mathbf{C}}^{n\times n} and M\in {\mathbf{C}}^{n\times n} be Hermitian and skewHermitian matrices, respectively. By using Theorem 2, the following conclusions will be obtained immediately.
Corollary 1 Let N\in {\mathbf{C}}^{n\times n} and M\in {\mathbf{C}}^{n\times n} be Hermitian and skewHermitian matrices, respectively. The following matrix trace inequalities are satisfied.

(I)
If H\ge 0, then
tr\left({({(N+M)}^{\ast}(N+M))}^{r}\right)\ge tr\left({N}^{2r}\right). 
(II)
If H>0, then
tr\left({({(N+M)}^{\ast}(N+M))}^{r}\right)\le tr\left({N}^{2r}\right). 
(III)
tr\left({({(N+M)}^{\ast}(N+M))}^{\frac{2k+1}{2}}\right)\ge tr\left({N}^{2k+1}\right).
Based on Corollary 1, the following results can be obtained when r=\frac{1}{2}. This implies that the proof of the general case of Lemma 1 is completed.
Corollary 2 Let N\in {\mathbf{C}}^{n\times n} and M\in {\mathbf{C}}^{n\times n} be a Hermitian and skewHermitian matrices, respectively. The following matrix trace inequalities are satisfied.

(I)
If N\ge 0, then
tr\left({({(N+M)}^{\ast}(N+M))}^{\frac{1}{2}}\right)\ge tr(N). 
(II)
If N>0, then
tr\left({({(N+M)}^{\ast}(N+M))}^{\frac{1}{2}}\right)\le tr\left({N}^{1}\right).
4 Conclusions
This paper proves several matrix trace inequalities on Hermitian and skewHermitian matrices. These matrix trace inequalities can be applied to design and analyze interiorpoint methods (IPMs) for semidefinite optimization (SDO). In addition, matrix trace inequalities have many potential applications in control theory, for example, stabilization of timedelay systems (see [15–18]).
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Acknowledgements
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.
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Gao, X., Wang, G., Zhang, X. et al. Several matrix trace inequalities on Hermitian and skewHermitian matrices. J Inequal Appl 2014, 358 (2014). https://doi.org/10.1186/1029242X2014358
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DOI: https://doi.org/10.1186/1029242X2014358
Keywords
 Hermitian matrix
 skewHermitian matrix
 Hermitian positive definite matrix
 semidefinite optimization
 interiorpoint methods