- Research
- Open Access

# Inequalities for eigenvalues of matrices

- Xiaozeng Xu
^{1, 2}Email author and - Chuanjiang He
^{1}

**2013**:6

https://doi.org/10.1186/1029-242X-2013-6

© Xu and He; licensee Springer 2013

**Received:**5 July 2012**Accepted:**14 December 2012**Published:**4 January 2013

## Abstract

The purpose of the paper is to present some inequalities for eigenvalues of positive semidefinite matrices.

**MSC:**15A18, 15A60.

## Keywords

- singular values
- eigenvalues
- unitarily invariant norm

## 1 Introduction

Throughout this paper, ${M}_{n}$ denotes the space of $n\times n$ complex matrices and ${H}_{n}$ denotes the set of all Hermitian matrices in ${M}_{n}$. Let $A,B\in {H}_{n}$; the order relation $A\ge B$ means, as usual, that $A-B$ is positive semidefinite. We always denote the singular values of *A* by ${s}_{1}(A)\ge \cdots \ge {s}_{n}(A)$. If *A* has real eigenvalues, we label them as ${\lambda}_{1}(A)\ge \cdots \ge {\lambda}_{n}(A)$. Let $\parallel \cdot \parallel $ denote any unitarily invariant norm on ${M}_{n}$. We denote by $|A|$ the absolute value operator of *A*, that is, $|A|={({A}^{\ast}A)}^{\frac{1}{2}}$, where ${A}^{\ast}$ is the adjoint operator of *A*.

*a*,

*b*, the arithmetic-geometric mean inequality says that

This is a matrix version of (1.1). For more information on matrix versions of the arithmetic-geometric mean inequality, the reader is referred to [1–11] and the references therein.

As pointed out in [10], p.198], although the arithmetic-geometric mean inequalities can be written in different ways and each of them may be obtained from the other, the matrix versions suggested by them are different.

In this note, we obtain a refinement of (1.2) and a log-majorization inequality for eigenvalues. As an application of our result, we give a matrix version of (1.3).

## 2 Main results

We begin this section with the following lemma, which is a question posed by Bhatia and Kittaneh [1] (see also [8, 10]) and settled in the affirmative by Drury in [2].

**Lemma 2.1**

*Let*$A,B\in {M}_{n}$

*be positive semidefinite*.

*Then*

*As a consequence of Lemma*2.1,

*we have*

*It is a matrix version of the arithmetic*-

*geometric mean inequality*.

*By properties of the matrix square function*,

*we know that this last inequality is stronger than the assertion*

*which is due to Bhatia and Kittaneh* [1]*and is also a matrix version of* (1.1).

**Theorem 2.1**

*Let*$A,B\in {M}_{n}$

*be positive semidefinite*.

*Then for all*$m=1,2,\dots $ ,

*Proof*By Lemma 2.1, we have

*A*,

*B*by ${A}^{1/2}$, ${B}^{1/2}$ in (2.3), we have

This completes the proof. □

**Remark 2.1**Let $A,B\in {M}_{n}$ be positive semidefinite. Note that

Therefore, the inequality (2.2) is a refinement of the inequality (1.2).

**Remark 2.2**For $m=1$, by (1.2), we have

**Theorem 2.2**

*Let*$A,B\in {M}_{n}$

*be positive semidefinite*.

*Then*

*Proof*By Weyl’s inequality, Horn’s inequality and Lemma 2.1, we have

This completes the proof. □

**Remark 2.3**As an application of Theorem 2.2, we now present a matrix version of (1.3). Taking $v=\frac{1}{2}$ in this last inequality, we have

This is a matrix version of (1.3).

## Declarations

### Acknowledgements

The authors wish to express their heartfelt thanks to the referees for their detailed and helpful suggestions for revising the manuscript.

## Authors’ Affiliations

## References

- Bhatia R, Kittaneh F: Notes on matrix arithmetic-geometric mean inequalities.
*Linear Algebra Appl.*2000, 308: 203–211. 10.1016/S0024-3795(00)00048-3MATHMathSciNetView ArticleGoogle Scholar - Drury SW: On a question of Bhatia and Kittaneh.
*Linear Algebra Appl.*2012, 437: 1955–1960. 10.1016/j.laa.2012.04.040MATHMathSciNetView ArticleGoogle Scholar - Audenaert KMR: A singular value inequality for Heinz means.
*Linear Algebra Appl.*2007, 422: 279–283. 10.1016/j.laa.2006.10.006MATHMathSciNetView ArticleGoogle Scholar - Bhatia R:
*Matrix Analysis*. Springer, New York; 1997.View ArticleGoogle Scholar - Bhatia R, Davis C: More matrix forms of the arithmetic-geometric mean inequality.
*SIAM J. Matrix Anal. Appl.*1993, 14: 132–136. 10.1137/0614012MATHMathSciNetView ArticleGoogle Scholar - Bhatia R, Kittaneh F: On the singular values of a product of operators.
*SIAM J. Matrix Anal. Appl.*1990, 11: 272–277. 10.1137/0611018MATHMathSciNetView ArticleGoogle Scholar - Bhatia R: Interpolating the arithmetic-geometric mean inequality and its operator version.
*Linear Algebra Appl.*2006, 413: 355–363. 10.1016/j.laa.2005.03.005MATHMathSciNetView ArticleGoogle Scholar - Bhatia R, Kittaneh F: The matrix arithmetic-geometric mean inequality revisited.
*Linear Algebra Appl.*2008, 428: 2177–2191. 10.1016/j.laa.2007.11.030MATHMathSciNetView ArticleGoogle Scholar - Kosaki H: Arithmetic-geometric mean and related inequalities for operators.
*J. Funct. Anal.*1998, 156: 429–451. 10.1006/jfan.1998.3258MATHMathSciNetView ArticleGoogle Scholar - Bhatia R:
*Positive Definite Matrices*. Princeton University Press, Princeton; 2007.Google Scholar - Zhan X Lecture Notes in Mathematics 1790. In
*Matrix Inequalities*. Springer, Berlin; 2002.View ArticleGoogle Scholar - Ando T: Matrix Young inequalities. Oper. Theory Adv. Appl. 75.
*Operator Theory in Function Spaces and Banach Lattices*1995, 33–38.View ArticleGoogle Scholar - Tao Y: More results on singular value inequalities of matrices.
*Linear Algebra Appl.*2006, 416: 724–729. 10.1016/j.laa.2005.12.017MATHMathSciNetView ArticleGoogle Scholar - Araki H: On an inequality of Lieb and Thirring.
*Lett. Math. Phys.*1990, 19: 167–170. 10.1007/BF01045887MATHMathSciNetView ArticleGoogle Scholar - Hiai F: Matrix analysis: matrix monotone functions, matrix means, and majorization.
*Interdiscip. Inf. Sci.*2010, 16: 139–248.MATHMathSciNetGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.