Positive Linear Maps and Perturbation bounds of Matrices

TL;DR

This paper explores how positive unital linear maps can be utilized to derive bounds on the eigenvalue differences, spread, and condition number of matrices, providing new insights into matrix perturbation analysis.

Contribution

It introduces novel bounds for eigenvalue distances and matrix properties using positive unital linear maps, advancing matrix perturbation theory.

Findings

Lower bounds for eigenvalue differences of normal matrices

Bounds for the spread of Hermitian matrices

Estimates for matrix condition numbers

Abstract

We show how positive unital linear maps can be used to obtain lower bounds for the maximum distance between the eigenvalues of two normal matrices. Some related bounds for the spread and condition number of Hermitian matrices are also discussed here.