Harmonic analysis for rank-1 Randomised Horn Problems

TL;DR

This paper extends harmonic analysis techniques to derive unified analytical results for rank-1 perturbations in various randomised Horn problems involving Hermitian and unitary matrices, enhancing understanding of their spectral properties.

Contribution

It generalizes existing results to rank-1 perturbations of products of positive definite Hermitian matrices using harmonic analysis and spherical transforms, providing a unified proof framework.

Findings

Unified analytical results for rank-1 perturbations in Horn problems

Development of spherical transform on the unitary group and proof of invertibility

Extension of harmonic analysis methods to new matrix group cases

Abstract

The randomised Horn problem, in both its additive and multiplicative version, has recently drawn increasing interest. Especially, closed analytical results have been found for the rank-1 perturbation of sums of Hermitian matrices and products of unitary matrices. We will generalise these results to rank-1 perturbations for products of positive definite Hermitian matrices and prove the other results in a new unified way. Our ideas work along harmonic analysis for matrix groups via spherical transforms that have been successfully applied in products of random matrices in the past years. In order to achieve the unified derivation of all three cases, we define the spherical transform on the unitary group and prove its invertibility.