An informal introduction to perturbations of matrices determined up to similarity or congruence

TL;DR

This paper surveys how canonical forms of matrices under similarity, congruence, and *congruence change discontinuously with perturbations, and introduces miniversal deformations as stable normal forms nearby.

Contribution

It provides an overview of the behavior of matrix canonical forms under perturbations and introduces the concept of miniversal deformations as stable normal forms.

Findings

Canonical forms depend discontinuously on matrix entries.

Miniversal deformations depend continuously on perturbations.

Normal forms provide stable approximations near a given matrix.

Abstract

The reductions of a square complex matrix A to its canonical forms under transformations of similarity, congruence, or *congruence are unstable operations: these canonical forms and reduction transformations depend discontinuously on the entries of A. We survey results about their behavior under perturbations of A and about normal forms of all matrices A+E in a neighborhood of A with respect to similarity, congruence, or *congruence. These normal forms are called miniversal deformations of A; they are not uniquely determined by A+E, but they are simple and depend continuously on the entries of E.