Canonical Form for Matrices Under Unitary Congruence Transformations. I: Conjugate-normal Matrices

Abstract

A unitary congruence transform of a given square matrix A is $UA\tilde U$, where U is a unitary matrix, and $\tilde U$ is its transpose. The problem of finding the canonical (i.e., the simplest) matrix $A^0 $ in the set of all unitary congruence transforms of A is solved for matrices which satisfy $AA^ + = ( {A^ + A} )^ * $, $A^ + $ and $A^ * $ being the adjoint and the complex conjugate of A. Such matrices are called conjugate-normal. Symmetric, skew-symmetric and unitary matrices are special cases, but do not exhaust all conjugate-normal ones. The canonical form $A^ 0 $ turns out to be the direct sum of a diagonal matrix and of several $2 \times 2$ matrices of the form $( {\begin{array}{*{20}c} 0 & {| a |} \\ a & 0 \\ \end{array} } )$ . A particular matrix Z and the general form $Z'$ of U which transforms A into $A^0 $ by congruence are derived.