Spectral variation of normal matrices (Q1195341)

The spectral distance \(sd(A,B)\) between \(n\times n\) complex matrices \(A\) and \(B\) with the eigenvalues \(\alpha_ 1,\dots,\alpha_ n\) and \(\beta_ 1,\dots,\beta_ n\) is defined as the minimum of \(\max_ k|\alpha_ k-\beta_{\pi(k)}|\) over all permutations \(\pi\) of \(1,\dots,n\). For pairs of some special normal matrices, it has been known that the operator (spectral) norm \(| A-B|\) of \(A-B\) gives the upper bound for \(sd(A,B)\). After reviewing these results, the author systematically carries out numerical experiments designed to obtain for \(n=3\) the lower bound for \(c\) such that \(sd(A,B)\leq c| A-B|\) and explains how the known results summarized in the paper help to explain the extremal examples. It is reported that \(c>1\), so that the conjecture by \textit{L. Mirsky} [Q. J. Math. Oxf., II. Ser. 11, 50-59 (1960; Zbl 0105.011)] is not true.