Canonical Forms for Unitary Congruence and *Congruence

TL;DR

This paper develops canonical forms for complex matrices under unitary congruence and *congruence, extending known classifications and exploring new cases, while also proving certain classification problems are unitarily wild and unsolvable.

Contribution

It introduces systematic canonical forms for matrices with specific normality conditions and investigates the complexity of related classification problems.

Findings

Canonical forms for matrices with normal A^2 or ar{A}A.

Unified treatment of conjugate normal, involutory, and projection matrices.

Certain classification problems are unitarily wild and unsolvable.

Abstract

We use methods of the general theory of congruence and *congruence for complex matrices--regularization and cosquares-to determine a unitary congruence canonical form (respectively, a unitary *congruence canonical form) for complex matrices A such that \bar{A}A (respectively, A^2) is normal. As special cases of our canonical forms, we obtain-in a coherent and systematic way-known canonical forms for conjugate normal, congruence normal, coninvolutory, involutory, projection, and unitary matrices. But we also obtain canonical forms for matrices whose squares are Hermitian or normal, and other cases that do not seem to have been investigated previously. We show that the classification problems under (a) unitary *congruence when A^3 is normal, and (b) unitary congruence when A\bar{A}A is normal, are both unitarily wild, so there is no reasonable hope that a simple solution to them can be found.