Canonical Forms for Unitary Congruence and *Congruence

摘要

We use methods of the general theory of congruence and *congruence forcomplex matrices--regularization and cosquares-to determine a unitarycongruence 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 andsystematic way-known canonical forms for conjugate normal, congruence normal,coninvolutory, involutory, projection, and unitary matrices. But we also obtaincanonical forms for matrices whose squares are Hermitian or normal, and othercases that do not seem to have been investigated previously. We show that the classification problems under (a) unitary *congruence whenA^3 is normal, and (b) unitary congruence when A\bar{A}A is normal, are bothunitarily wild, so there is no reasonable hope that a simple solution to themcan be found.