On normal forms for the similarity classes of matrices and pairs of matrices

TL;DR

This paper investigates the structure of similarity classes of matrices and pairs of matrices, establishing a finite union of affine subspaces for single matrices over any field, but showing such a structure does not extend to pairs over algebraically closed fields of characteristic zero.

Contribution

It provides a definitive answer to two longstanding questions about the geometric structure of similarity classes of matrices and pairs of matrices.

Findings

Finite union of affine subspaces for similarity classes of matrices over any field.

Failure of analogous structure for pairs of matrices over algebraically closed fields of characteristic zero.

Clarification of the geometric and algebraic properties of matrix similarity classes.

Abstract

We answer two questions posed 1998 in the book 'Arnolds problems'. First, over any field k there is a representative system for the similarity classes of nxn-matrices which is a finite disjoint union of affine subspaces. And second, for n>1 an analogous statement fails for pairs of nxn-matrices over any algebraically closed field of characteristic 0.