Simultaneous similarity and triangularization of sets of 2 by 2 matrices

TL;DR

This paper characterizes when sets of 2x2 matrices are simultaneously triangularizable, providing criteria, canonical forms, and invariants, with implications for understanding matrix similarity and triangularization over various fields.

Contribution

It establishes a criterion for simultaneous triangularization of 2x2 matrices, introduces a minimal invariant map for classifying sequences, and describes canonical forms over algebraically closed fields.

Findings

A necessary and sufficient condition for simultaneous triangularization.

A simple numerical criterion for triangularization.

Canonical forms and invariants for sequences of 2x2 matrices.

Abstract

Let $\mathcal{A}=(A_{1},...,A_{n},...)$ be a finite or infinite sequence of $2\times2$ matrices with entries in an integral domain. We show that, except for a very special case, $\mathcal{A}$ is (simultaneously) triangularizable if and only if all pairs $(A_{j},A_{k})$ are triangularizable, for $1\leq j,k\leq\infty$. We also provide a simple numerical criterion for triangularization. Using constructive methods in invariant theory, we define a map (with the minimal number of invariants) that distinguishes simultaneous similarity classes for non-commutative sequences over a field of characteristic $\neq2$. We also describe canonical forms for sequences of $2\times2$ matrices over algebraically closed fields, and give a method for finding sequences with a given set of invariants.