Harmonic analysis for rank-1 Randomised Horn Problems

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.

24 pages

References in corpus (5)

• Quantum marginal problem and representations of the symmetric group
• On Horn's Problem and its Volume Function
• Co-rank 1 projections and the randomised Horn problem
• Duistermaat-Heckman measure and the mixture of quantum states
• Products of Complex Rectangular and Hermitian Random Matrices

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• One-matrix differential reformulation of two-matrix models