Multiplication law and S transform for non-hermitian random matrices

TL;DR

This paper introduces a multiplication law and a generalized S transform for non-hermitian random matrices, enabling easier analysis of their eigenvalue distributions and extending classical methods to centered ensembles.

Contribution

It develops a multiplication law and a non-hermitian S transform, generalizing existing tools to handle non-hermitian matrices and centered ensembles.

Findings

Derived a multiplication law for non-hermitian matrices

Defined a non-hermitian S transform extending Voiculescu's approach

Extended classical S transform to centered ensembles

Abstract

We derive a multiplication law for free non-hermitian random matrices allowing for an easy reconstruction of the two-dimensional eigenvalue distribution of the product ensemble from the characteristics of the individual ensembles. We define the corresponding non-hermitian S transform being a natural generalization of the Voiculescu S transform. In addition we extend the classical hermitian S transform approach to deal with the situation when the random matrix ensemble factors have vanishing mean including the case when both of them are centered. We use planar diagrammatic techniques to derive these results.