- Open Access
Inequalities for eigenvalues of matrices
© Xu and He; licensee Springer 2013
Received: 5 July 2012
Accepted: 14 December 2012
Published: 4 January 2013
The purpose of the paper is to present some inequalities for eigenvalues of positive semidefinite matrices.
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.
As pointed out in , 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
which is due to Bhatia and Kittaneh and is also a matrix version of (1.1).
This completes the proof. □
Therefore, the inequality (2.2) is a refinement of the inequality (1.2).
This completes the proof. □
This is a matrix version of (1.3).
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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