Aggregate Bounds on the eigenvalues of the principal submatrices of a Hermitian matrix and majorization relations
Hristo Sendov, Mengxu Yuan
TL;DR
The paper addresses how the eigenvalues of an $n\times n$ Hermitian matrix relate to those of all its $(n-1)\times(n-1)$ and, more generally, $m\times m$ principal submatrices. It introduces a weighted polynomial-zeros framework to derive sharp, aggregate bounds on sums of consecutive zeros, extending Thompson’s bounds, and demonstrates that these bounds induce majorization relations among the spectra of principal submatrices. The authors establish a general theorem (renormalized_bounds) for weighted bounds, prove equal-weight and general-weight cases, and derive majorization results that recover Schur’s classical majorization and Szász’s inequalities. They also provide geometric interpretations via eigenvalue compressions and show a spectral hierarchy across submatrix sizes, highlighting the broader impact on spectral inequalities and majorization theory. Overall, the work unifies polynomial-root interlacing with matrix spectral bounds to yield a comprehensive framework for principal-submatrix spectra.
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.