New properties for certain positive semidefinite matrices

TL;DR

This paper introduces new properties related to inequalities involving singular values and eigenvalues of block positive semidefinite matrices, exploring their interrelations with numerous illustrative examples.

Contribution

It presents novel notions and inequalities for 2x2 block positive semidefinite matrices, expanding understanding of their spectral properties.

Findings

New inequalities between singular values and eigenvalues

Relations between arithmetic and geometric means of blocks

Numerous illustrative examples

Abstract

We bring in some new notions associated with $2\times 2$ block positive semidefinite matrices. These notions concern the inequalities between the singular values of the off diagonal blocks and the eigenvalues of the arithmetic mean or geometric mean of the diagonal blocks. We investigate some relations between them. Many examples are included to illustrate these relations.