Off-diagonal submatrices of a Hermitian matrix

Abstract

A necessary and sufficient condition is given to a p × q p\times q complex matrix X X to be an off-diagonal block of an n × n n\times n Hermitian matrix C C with prescribed eigenvalues (in terms of the eigenvalues of C C and singular values of X X ). The proof depends on some recent breakthroughs in the study of spectral inequalities on the sum of Hermitian matrices by Klyachko and Fulton. Some interesting geometrical properties of the set S S of all such matrices are derived from the main result. These results improve earlier ones that only give partial information for the set S S .