On Normal and Binormal Matrices

Abstract

The problem discussed is how to obtain a normal matrix from a binormal one and, conversely, a binormal matrix from a normal one via the right multiplication on a suitable unitary matrix. Let $N$ be a normal matrix badly conditioned with respect to inversion, that is, having a large condition number $\textrm{cond}_{2}N$. We show that, among the binormal matrices $B$ that can be obtained from $N$, there is a matrix whose eigenvalues have individual condition numbers of order $(\textrm{cond}_{2}N)^{1/2}$.