Canonical Form for Matrices Under Unitary Congruence Transformations. II: Congruence-Normal Matrices

Abstract

A square matrix A for which $B = AA^ * $ is normal is called a congruence-normal matrix. The canonical form $A^0 $ of A is derived by a unitary congruence transformation $UA\tilde U$, using properties of commuting auxiliary matrices B, $C = \tilde AA^ * $ and $D = AA^ + $ (where $\tilde A,A^ * $ and $A^ + $ are respectively the transpose, the complex conjugate and the adjoint of the matrix A). It is found that $A^0 $ is the direct sum of a diagonal part and two by two submatrices of the form $( {\begin{array}{*{20}c} 0 & a \\ b & 0 \\ \end{array} } )$ where $a\geqq 0$, and $b \ne 0$ has a nonnegative imaginary part. Various fields of application are indicated, such as electrical network theory, self-consistent calculations in superconducting systems, linear-antilinear representations of groups, etc.