Isospectrality and matrices with concentric circular higher rank numerical ranges

TL;DR

This paper characterizes conditions under which pairs of Hermitian matrices produce spectra independent of a parameter, linking isospectrality to higher rank numerical ranges and solutions to Lax's equation.

Contribution

It provides a characterization of isospectral Hermitian matrix pairs using higher rank numerical ranges and Lax's equation solutions.

Findings

Spectrum of the trigonometric matrix pencil is independent of t under certain conditions.

First higher rank numerical ranges are circular disks centered at 0.

Unitary similarity involves solving Lax's equation.

Abstract

We characterize under what conditions $n\times n$ Hermitian matrices $A_1$ and $A_2$ have the property that the spectrum of $\cos t A_1 + \sin t A_2$ is independent of $t$ (thus, the trigonometric pencil $\cos t A_1 + \sin t A_2$ is isospectral). One of the characterizations requires the first $\lceil \frac{n}{2} \rceil$ higher rank numerical ranges of the matrix $A_1+iA_2$ to be circular disks with center 0. Finding the unitary similarity between $\cos t A_1 + \sin t A_2$ and, say, $A_1$ involves finding a solution to Lax's equation.