About the logarithm function over the matrices

TL;DR

This paper investigates conditions under which the matrix logarithm satisfies the additive property and shows that, under certain conditions, the matrices involved must commute, linking matrix analysis with complex analysis techniques.

Contribution

It establishes new conditions for the logarithm of matrix products to be additive, specifically for 2x2 matrices or simultaneously triangularizable matrices, connecting matrix properties with complex analysis.

Findings

For 2x2 matrices, the matrices commute if the logarithm property holds.

For higher dimensions, simultaneous triangularizability implies commutativity under the logarithm condition.

The problem is reduced to complex analysis in both cases.

Abstract

We prove the following results: let x,y be (n,n) complex matrices such that x,y,xy have no eigenvalue in ]-infinity,0] and log(xy)=log(x)+log(y). If n=2, or if n>2 and x,y are simultaneously triangularizable, then x,y commute. In both cases we reduce the problem to a result in complex analysis.