The generic canonical form for $^\star$congruence of matrices

TL;DR

This paper establishes the generic canonical forms for complex matrices under congruence and *-congruence, revealing a fundamental difference in the number of forms needed to represent typical matrices.

Contribution

It provides the first comprehensive characterization of generic canonical forms under congruence and *-congruence for complex matrices, highlighting a key dichotomy between these relations.

Findings

The set of complex matrices is the closure of an open subset with a unique canonical form under congruence.

The set of complex matrices is the union of closures of several open subsets with specific *-congruence canonical forms.

There is only one generic canonical form under congruence, but multiple under *-congruence, depending on matrix size.

Abstract

First, we prove that the set of $n\times n$ complex matrices is the closure of a certain open subset whose elements have a very specific canonical form under congruence, which is uniquely determined up to the values of some parameters, but which has a slightly different expression depending on whether $n$ is even or odd. As a consequence, the canonical form under congruence of the elements of this subset can be considered the generic canonical form under congruence of complex $n\times n$ matrices. Second, we prove that the set of $n\times n$ complex matrices is the union of the closures of certain $\lfloor n/2\rfloor+1$ open subsets and that, for each of these subsets, its elements have a very specific canonical form under $^*$congruence, which is uniquely determined up to the values of some parameters. As a consequence, the $\lfloor n/2\rfloor+1$ canonical forms under $^*$congruence of the elements of each of these subsets can be considered the generic canonical forms under $^*$congruence of complex $n\times n$ matrices. So, there is only one generic canonical form under congruence whereas the number of generic canonical forms under $^*$congruence is $\lfloor n/2\rfloor+1$ instead, which reveals a strong dichotomy between the relations of congruence and $^*$congruence with respect to generic structures. In other words, we determine in this paper the generic matrix representations of $n\times n$ bilinear and sesquilinear forms in $\mathbb{C}^n \times \mathbb{C}^n$.