Asymptotic metrics on the space of matrices under the commutation relation

TL;DR

This paper investigates how the norm of the commutator behaves as an asymptotic metric on the space of matrices, showing it satisfies the triangle inequality approximately for large matrices from certain distributions.

Contribution

It demonstrates that the commutator norm acts as an almost metric on the quotient space of commuting matrices in the asymptotic limit.

Findings

Commutator norm satisfies the triangle inequality asymptotically for large matrices.

The semi-metric property holds for matrices drawn from a 'good' distribution.

The results apply to the quotient space of commuting matrices.

Abstract

We show that the norm of the commutator defines "almost a metric" on the quotient space of commuting matrices, in the sense that it is a semi-metric satisfying the triangle inequality asymptotically for large matrices drawn from a "good" distribution.