Simultaneous similarity and triangularization of sets of 2 by 2 matrices

Details Simultaneous similarity and triangularization of sets of 2 by 2 matrices Simultaneous similarity and triangularization of sets of 2 by 2 matrices

2008-09-17

arxiv.org/abs/0809.3032v1

Mathematics

Rings and Algebras

22 pages

Scientific paper

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.

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