Unitary congruence to a conjugate-normal matrix

Abstract

A matrix A ∈ M n (C) is said to be conjugate-normal if $ A{A^{*}}=\overline{{{A^{*}}A}} $ . The following result (which is the congruence analog of a recent result by T. G. Gerasimova) is proved: A matrix B ∈ M n (C) is unitarily congruent to a conjugate-normal matrix A of and only if $$ \begin{array}{*{20}{c}} {\mathrm{tr}\left[ {{{{\left( {\bar{A}A} \right)}}^i}} \right]=\mathrm{tr}\left[ {{{{\left( {\bar{B}B} \right)}}^i}} \right],} & {i=1,\ldots,n,} \end{array} $$ and $$ {{\left\| A \right\|}_F}={{\left\| B \right\|}_F}. $$ This result dramatically reduce the amount of computational work for verifying unitary congruence as compared to the case of general matrices A and B. Bibliography: 8 titles.