A Characterization on Singular Value Inequalities of Matrices

Abstract

We obtain a characterization of pair matrices A and B of order n such that where denotes the j-th largest singular values of It can imply not only some well-known singular value inequalities for sums and direct sums of matrices but also Zhan’s result related to singular values of differences of positive semidefinite matrices. In addition, some related and new inequalities are also obtained.

1. Introduction

Let denote the vector space of all complex matrices, and let be the set of all Hermitian matrices of order . We always denote the eigenvalues of in decreasing order by . For , we use the notation or to mean that is positive semidefinite. Clearly, “” and “” define two partial orders on each of which is called Löwner partial order. In particular, (res., ) means that B is positive semidefinite (res., B is positive definite). For , the singular values of , denoted by , are the eigenvalues of the positive semidefinite matrix , enumerated as and repeated according to multiplicity. It follows that the singular values of a normal matrix are just the moduli of its eigenvalues. In particular, if is positive semidefinite, then singular values and eigenvalues of T are the same. For more information on this related topic, see [1–3]. Recall that a complex matrix is called contraction if , or equivalently where denotes the spectral norm, the identity matrix of order respectively.

Here, we denote the block matrix by .

This paper intends to give a characterization of pair matrices such that . It can generalize some singular value inequalities for sums and direct sums of matrices due to Hirzallah and Kittaneh [4]. Using this characterization, we give a new proof of Zhan’s result related to singular values of differences of positive semidefinite matrices [5]. Several applications of this characterization are presented, and some related and new inequalities are also obtained.

2. Main Results

The following well-known results are due to Ky Fan.

Lemma 1. Let ThenIn particular,where denotes the spectral norm.

Next, we will prove the following useful result.

Theorem 1. Let . Then,if and only if contractive matrices W and V exist such that In particular, suppose A and B are positive semidefinite. Then,if and only if there exists a contractive matrix W such that

Proof. The sufficiency follows from Lemma 1. We prove the necessity.
Consider the singular value decompositions and . Then, unitary matrices , and exist such thatSettingDenote . It is clear that . Then,Let and . It is trivial that both W and V are contractive. Then, .
Let A and B be positive semidefinite. Note that for positive semidefinite matrices, singular values and eigenvalues are the same. Using the spectral decompositions of and , the unitary matrices and exist such thatThe left proof is similar to the above. This completes the proof.
It is clear that for ,where O is a zero square matrix of any order. An interesting consequence of Theorem 1 is the following.

Corollary 1 (see [4], Theorem 2.1). Let . Then,for

Proof. DenoteThen, . Then,where O is the zero matrix of order . It is clear that both and are contractive matrices. By Theorem 1, this completes the proof.
The following result generalizes the result due to Hirzallah and Kittaneh ([4], Corollary 2.1).

Theorem 2. Let , and let be complex numbers such that Then, for

Proof. Note that each for some . DenoteThen,with contractive matrices . By Theorem 1, this completes the proof.

Remark. This result admits the following important special cases:(1)Let , and let be nonnegative real numbers with . Then,(2)Let . Then,In particular, letting , we havewhich can be regarded as a complement to Hirzallah and Kittaneh’s inequality (see ([4], Corollary 2.3)), which sayswhere and denote the real part and the imaginary part of , respectively.
Applying Theorem 2, we have the following result.

Corollary 2. Let . Then,

Proof. Considering the Cartesian decomposition of . By Theorem 2, we have for ,The last equality on the above is due to the fact that singular values are unitarily invariant: , for every A and all unitary .

Remark. Next, we show that equality is possible in inequality (20) for some nonzero square matrix T. Consider the matrix . Then,It is clear that .
The well-known arithmetic-geometric mean inequality for singular values due to Bhatia and Kittaneh [6] says that if , thenZhan [5] asserts that for , if and , thenIn [7], Tao has proved that if such that , thenIt has been pointed out [7] that the three inequalities (23)–(25) are equivalent.
Recently, Audeh and Kittaneh [8] proved that let If , thenNext, we will interpolate Audeh and Kittaneh’s inequality (26) by proving.

Theorem 3. Let such that . Then,

Proof. Since , we havefor some . Then,Note that singular values and eigenvalues of positive semidefinite matrices are the same. By Theorem 1.27 in [3], for , thenAppling Theorem 2 and Theorem 1.27 in [3], we obtain that for ,i.e.,Combining inequality (25) by Tao and the above inequality gives (27).
Let be the spectral decomposition of with U unitary and . DenoteThen , , and . This is called the Jordan decomposition of As is known, there are at least four different proofs of Zhan’s result [5, 9]. Applying Theorem 1 and the Jordan decomposition, we give another direct proof of Zhan’s inequality (8).

Proof. Denote . Considering the Jordan decomposition of , . Recall from ([3], Theorem 3.6) that if with thenBy Theorem 1, there exist two contractive matrices and such thatDenote . Then, is a contractive matrix and so . For ,where . By Theorem 1, this completes the proof. [10].

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare there are no conflicts of interest.

Authors’ Contributions

The authors read and approved the final manuscript.

Acknowledgments

This work was supported by the Anhui Provincial Natural Science Foundation (1708085QA05) and the Key Program in the Youth Elite Support Plan in Universities of Anhui Province (gxyqZD2018047).