Eigenvalues and diagonal elements.

A basic theorem in linear algebra says that if the eigenvalues and the diagonal entries of a Hermitian matrix A are ordered as λ 1 ≤ λ 2 ≤... ≤ λ n and a 1 ≤ a 2 ≤... ≤ a n , respectively, then λ 1 ≤ a 1 . We show that for some special classes of Hermitian matrices this can be extended to inequalities of the form λ k ≤ a 2 k - 1 , k = 1 , 2 ,... , ⌈ n 2 ⌉ . [ABSTRACT FROM AUTHOR]