Eigenvalues and diagonal elements

Abstract

A basic theorem in linear algebra says that if the eigenvalues and the diagonal entries of a Hermitian matrix A are ordered as $\lambda _{1}\le \lambda _{2}\le ...\le \lambda _{n}$ and $a_{1}\le a_{2}\le ...\le a_{n}$, respectively, then $\lambda _{1}\le a_{1}$. We show that for some special classes of Hermitian matrices this can be extended to inequalities of the form $\lambda _{k}\le a_{2k-1}$, $k=1,2,...,\lceil \frac{n}{2}\rceil$.