Canonical Forms for Unitary Congruence and *Congruence

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.

43 pages, to be published in Linear Multilinear Algebra

References in corpus (5)

• Canonical forms for complex matrix congruence and *congruence
• Complexity of matrix problems
• Congruences of a square matrix and its transpose
• A regularization algorithm for matrices of bilinear and sesquilinear forms
• Unitary and Euclidean representations of a quiver

Cited by in corpus (6)

• Geometric picture of generalized-CP and Higgs-family transformations in the two-Higgs-doublet model
• Canonical matrices of bilinear and sesquilinear forms
• Tensor space representations of Temperley-Lieb algebra and generalized permutation matrices
• Simultaneous block diagonalization of a set of symmetric matrices via congruence
• A note on the singular value decomposition of (skew-)involutory and (skew-)coninvolutory matrices
• Classification of squared normal operators on unitary and Euclidean spaces