What about A,B if AB-BA and A commute

TL;DR

This paper investigates the properties of complex matrices A and B under specific conditions, showing they are simultaneously triangularizable in 2x2 cases but not necessarily for larger matrices, revealing limitations in matrix commutativity.

Contribution

It demonstrates that for 2x2 matrices, A and B are always simultaneously triangularizable when AB-BA and A commute, but for larger matrices, such pairs may not satisfy property L.

Findings

In 2x2 case, A and B are always simultaneously triangularizable.

For matrices of size 3 or more, counterexamples exist where A and B are not simultaneously triangularizable.

The property L of Motzkin-Taussky does not hold universally for larger matrices under these conditions.

Abstract

Let A,B be complex n,n complex matrices such that AB-BA and A commute. We show that, if n=2 then A,B are simultaneously triangularizable and if n>=3 then there exists such a couple A,B such that the pair (A,B) has not property L of Motzkin-Taussky and such that B and C are not simultaneously triangularizable.