The spectral spread of Hermitian matrices

TL;DR

This paper investigates the spectral spread of Hermitian matrices, introducing new inequalities and properties related to eigenvalue dispersion, with applications in matrix approximation and matrix analysis.

Contribution

It develops novel inequalities and extremal properties for the spectral spread of Hermitian matrices, extending existing results in matrix theory and eigenvalue analysis.

Findings

Derived inequalities related to Tao's inequality for anti-diagonal blocks

Established bounds for singular values of differences of positive semidefinite matrices

Explored extremal properties of direct rotations and distances in unitary orbits

Abstract

Let $A$ be a $n\times n$ complex Hermitian matrix and let $\lambda(A)=(\lambda_1,\ldots,\lambda_n)\in \mathbb{R}^n$ denote the eigenvalues of $A$, counting multiplicities and arranged in non-increasing order. Motivated by problems arising in the theory of low rank matrix approximation, we study the spectral spread of $A$, denoted $\text{Spr}^+(A)$, given by $\text{Spr}^+(A) =(\lambda_1-\lambda_{n}\, , \, \lambda_2-\lambda_{n-1},\ldots, \lambda_{k}-\lambda_{n-k+1})\in \mathbb{R}^k$, where $k=[n/2]$ (integer part). The spectral spread is a vector-valued measure of dispersion of the spectrum of $A$, that allows one to obtain several submajorization inequalities. In the present work we obtain inequalities that are related to Tao's inequality for anti-diagonal blocks of positive semidefinite matrices, Zhan's inequalities for the singular values of differences of positive semidefinite matrices, extremal properties of direct rotations between subspaces, generalized commutators and distances between matrices in the unitary orbit of a Hermitian matrix.