Change of the *congruence canonical form of 2-by-2 matrices under perturbations

TL;DR

This paper investigates how minor perturbations affect the *congruence canonical form of 2-by-2 matrices, providing a detailed classification and the structure of class closures.

Contribution

It introduces a comprehensive analysis of the stability of *congruence canonical forms under perturbations and constructs the Hasse diagram for class closures.

Findings

Perturbations can cause transitions between *congruence classes.

The Hasse diagram illustrates the closure relations among classes.

Canonical forms are sensitive to small matrix changes.

Abstract

We study how small perturbations of a 2-by-2 complex matrix can change its canonical form for *congruence. We construct the Hasse diagram for the closure ordering on the set of *congruence classes of 2-by-2 matrices.