A bound below for the convex hull of the spectrum of a matrix

TL;DR

This paper establishes a new geometric bound for the convex hull of a matrix's spectrum, using quadratic forms and ellipses related to the matrix's traceless part.

Contribution

It introduces a novel bound involving an ellipse determined by the quadratic form on traceless matrices, providing insight into the spectral convex hull.

Findings

Ellipse contained in the convex hull of the spectrum

Bound relates to the quadratic form Q(A)

Provides geometric insight into spectral properties

Abstract

In this note the following is shown. Consider the quadratic form on (complex) matrices Q(A):=tr(A^2). Let A be such a matrix. Then an ellipse can be found, with the vector from center to focus determined by the value of Q at the traceless part of A, which must be contained in the convex hull of the spectrum of A.