Traces of Matrix Products

Abstract

Given two noncommuting matrices, A and B, it is well known that AB and BA have the same trace. This extends to cyclic permutations of products of Aâs and Bâs. It is shown here that for 2Ã2 matrices A and B, whose elements are independent random variables with standard normal distributions, the probability that Tr(ABAB) > Tr(AB) is exactly 1/\sqrt{2} .