Aggregate Bounds on the eigenvalues of the principal submatrices of a Hermitian matrix and majorization relations

TL;DR

This paper extends classical bounds on eigenvalues of principal submatrices of Hermitian matrices, introduces stronger bounds via polynomial derivatives, and establishes majorization relations that encompass known inequalities.

Contribution

It provides stronger eigenvalue bounds for principal submatrices and derives new majorization relations, unifying and extending classical results.

Findings

Stronger bounds on eigenvalues of principal submatrices.

Majorization relations between eigenvalues of different-sized principal matrices.

Unified proof of classical inequalities like Schur and Szasz's inequalities.

Abstract

We extend bounds, proved by R.C. Thompson in 1966, on the sum of the $j$-th largest eigenvalues of the $(n-1) \times (n-1)$ principal matrices of an $n \times n$ Hermitian matrix. Our bounds are stronger than just summing up Thompson's bounds. We achieve the extensions as a corollary of a more general result giving bounds on the zeros of the generalized derivatives of polynomials with real roots. We use the extended bounds to obtain majorization relationships between the eigenvalues of all $m \times m$ principal matrices of an $n \times n$ Hermitian matrix. These majorization relationships imply both a well-known majorization result by Schur and the well-known Szasz's inequalities.