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Inequalities for eigenvalues of matrices
Journal of Inequalities and Applications volume 2013, Article number: 6 (2013)
Abstract
The purpose of the paper is to present some inequalities for eigenvalues of positive semidefinite matrices.
MSC:15A18, 15A60.
1 Introduction
Throughout this paper, denotes the space of complex matrices and denotes the set of all Hermitian matrices in . Let ; the order relation means, as usual, that is positive semidefinite. We always denote the singular values of A by . If A has real eigenvalues, we label them as . Let denote any unitarily invariant norm on . We denote by the absolute value operator of A, that is, , where is the adjoint operator of A.
For positive real number a, b, the arithmetic-geometric mean inequality says that
It is equivalent to
Let be positive semidefinite. Bhatia and Kittaneh [1] proved that for all ,
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.
It is easy to see that the arithmetic-geometric mean inequality is also equivalent to
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 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 be positive semidefinite. Then for all ,
Proof By Lemma 2.1, we have
Replacing A, B by , in (2.3), we have
This completes the proof. □
Remark 2.1 Let be positive semidefinite. Note that
Therefore, the inequality (2.2) is a refinement of the inequality (1.2).
Remark 2.2 For , by (1.2), we have
For , by (2.2), we have
In view of the inequalities (2.4) and (2.5), one may ask whether it is true that
for all . The answer is no. For , the inequality (2.6) is refuted by the following example:
Theorem 2.2 Let be positive semidefinite. Then
Proof By Weyl’s inequality, Horn’s inequality and Lemma 2.1, we have
Putting
in (2.7) gives
In response to a conjecture by Zhan [11], Audenaert [3] proved that if , then
The special case where was obtained earlier in [6, 12] and the special case where was obtained earlier in [13]. It follows from (2.8) and (2.9) that
This completes the proof. □
Remark 2.3 As an application of Theorem 2.2, we now present a matrix version of (1.3). Taking in this last inequality, we have
and so
which is equivalent to
Since weak log-majorization is stronger than weak majorization, we have
By Fan’s dominance theorem [4], p.93], we get
This is a matrix version of (1.3).
Next, we give another proof of the inequality (2.10). Araki [14] (also see [15]) obtained the following log-majorization inequality:
Putting
in (2.11) gives
and so
By Fan’s dominance theorem [4], p.93], we get
It follows from (2.1) and (2.12) that
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The authors wish to express their heartfelt thanks to the referees for their detailed and helpful suggestions for revising the manuscript.
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Xu, X., He, C. Inequalities for eigenvalues of matrices. J Inequal Appl 2013, 6 (2013). https://doi.org/10.1186/1029-242X-2013-6
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DOI: https://doi.org/10.1186/1029-242X-2013-6