Curves and envelopes that bound the spectrum of a matrix

TL;DR

This paper extends a method for bounding the eigenvalues of complex matrices by using multiple largest eigenvalues of their Hermitian parts, resulting in tighter spectral bounds and new geometric regions containing the spectrum.

Contribution

It generalizes previous eigenvalue bounding techniques by incorporating more eigenvalues, leading to improved inequalities and envelope regions for the spectrum of matrices.

Findings

Using three or more largest eigenvalues yields tighter spectral bounds.

New envelope regions are constructed that contain the matrix spectrum.

The method improves upon existing bounds by reducing the spectral region.

Abstract

A generalization of the method developed by Adam, Psarrakos and Tsatsomeros to find inequalities for the eigenvalues of a complex matrix A using knowledge of the largest eigenvalues of its Hermitian part H(A) is presented. The numerical range or field of values of A can be constructed as the intersection of half-planes determined by the largest eigenvalue of H($e^{i\theta}$A). Adam, Psarrakos and Tsatsomeros showed that using the two largest eigenvalues of H(A), the eigenvalues of A satisfy a cubic inequality and the envelope of such cubic curves defines a region in the complex plane smaller than the numerical range but still containing the spectrum of A. Here it is shown how using the three largest eigenvalues of H(A) or more, one obtains new inequalities for the eigenvalues of A and new envelope-type regions containing the spectrum of A.