Simultaneous similarity and triangularization of sets of 2 by 2 matrices

Let A=(A1,…,An,…) be a finite or infinite sequence of 2×2 matrices with entries in an integral domain. We show that, except in a very special case, A is (simultaneously) triangularizable if and only if all pairs (Aj,Ak) are triangularizable, for 1⩽j,k⩽∞. 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 ≠2. We also describe canonical forms for sequences of 2×2 matrices over algebraically closed fields, and give a method for finding sequences with a given set of invariants.