Isospectrality and matrices with concentric circular higher rank numerical ranges

Edward Poon; Hugo J. Woerdeman

doi:10.1016/j.laa.2021.08.025

paper

Isospectrality and matrices with concentric circular higher rank numerical ranges

arXiv:2103.12885 · doi:10.1016/j.laa.2021.08.025

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 $⌈ \frac{n}{2} ⌉$ higher rank numerical ranges of the matrix $A_{1} + i A_{2}$ 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.

Compared to the original submission, in the final example another solution P(t) is proposed as well as a solution for U(t)

Isospectrality and matrices with concentric circular higher rank numerical ranges

Abstract

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