Eigenvalue inequalities for positive block matrices with the inradius of the numerical range

TL;DR

This paper establishes eigenvalue inequalities for positive block matrices, highlighting the role of the inradius of the numerical range of off-diagonal blocks, and introduces related norm inequalities and a new conjecture.

Contribution

It provides novel eigenvalue bounds involving the inradius of the numerical range for block matrices, expanding understanding of their spectral properties.

Findings

Eigenvalue inequalities depend on the inradius of the numerical range.

Norm inequalities related to block matrices are derived.

A new conjecture on eigenvalue bounds is proposed.

Abstract

We prove some eigenvalue inequalities for positive semidefinite matrices partitioned into four blocks. The inradius of the numerical range of the off-diagonal block contributes to these estimates. Some related norm inequalities are given and a conjecture is proposed.