An eigenvalue majorization inequality for positive semidefinite block matrices

Abstract

Let be a Hermitian matrix. It is known that the vector of diagonal elements of H, diag(H), is majorized by the vector of the eigenvalues of H, λ(H), and that this majorization can be extended to the eigenvalues of diagonal blocks of H. Reverse majorization results for the eigenvalues are our goal. Under the additional assumptions that H is positive semidefinite and the block K is Hermitian, the main result of this article provides a reverse majorization inequality for the eigenvalues. This results in the following majorization inequalities when combined with known majorization inequalites on the left:

Keywords:

• majorization
• eigenvalue inequalities

AMS Subject Classifications::

• 15A42
• 06A06

Acknowledgements

The authors thank the anonymous referees for valuable suggestions that improved the presentation of this article. This research was supported by The Natural Sciences and Engineering Research Council of Canada.