Off-diagonal submatrices of a Hermitian matrix

Li, Chi-Kwong; Poon, Yiu-Tung

doi:10.1090/S0002-9939-04-07072-8

Off-diagonal submatrices of a Hermitian matrix
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by Chi-Kwong Li and Yiu-Tung Poon PDF

Proc. Amer. Math. Soc. 132 (2004), 2849-2856 Request permission

Abstract:

A necessary and sufficient condition is given to a $p\times q$ complex matrix $X$ to be an off-diagonal block of an $n\times n$ Hermitian matrix $C$ with prescribed eigenvalues (in terms of the eigenvalues of $C$ and singular values of $X$). 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 $S$ of all such matrices are derived from the main result. These results improve earlier ones that only give partial information for the set $S$.

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