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

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.