# Simultaneous similarity and triangularization of sets of 2 by 2 matrices - Mathematics > Rings and Algebras

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, exceptfor a very special case, $\mathcal{A}$ is simultaneously triangularizable ifand only if all pairs $A {j},A {k}$ are triangularizable, for $1\leqj,k\leq\infty$. We also provide a simple numerical criterion fortriangularization.Using constructive methods in invariant theory, we define a map with theminimal number of invariants that distinguishes simultaneous similarityclasses for non-commutative sequences over a field of characteristic $eq2$.We also describe canonical forms for sequences of $2\times2$ matrices overalgebraically closed fields, and give a method for finding sequences with agiven set of invariants.

Author: Carlos A. A. Florentino

Source: https://arxiv.org/