Note on Bounds for Eigenvalues using Traces

TL;DR

This paper demonstrates that many existing and new bounds on eigenvalues of complex matrices can be derived from inequalities related to the variance of real and complex numbers.

Contribution

It reveals that eigenvalue bounds are direct consequences of variance inequalities, unifying and simplifying their derivation.

Findings

Eigenvalue bounds follow from variance inequalities.

Unified approach simplifies understanding of eigenvalue bounds.

New bounds are derived from classical inequalities.

Abstract

We show that various old and new bounds involving eigenvalues of a complex n x n matrix are immediate consequences of the inequalities involving variance of real and complex numbers.