Vol. 3, No. 1, 2021

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Corank-1 projections and the randomised Horn problem

Peter J. Forrester and Jiyuan Zhang

Vol. 3 (2021), No. 1, 55–73
Abstract

Let x̂ be a normalised standard complex Gaussian vector, and project an Hermitian matrix A onto the hyperplane orthogonal to x̂. In a recent paper Faraut (Tunisian J. Math. 1 (2019), 585–606) has observed that the corresponding eigenvalue PDF has an almost identical structure to the eigenvalue PDF for the rank-1 perturbation A + bx̂x̂, and asks for an explanation. We provide one by way of a common derivation involving the secular equations and associated Jacobians. This applies also in a related setting, for example when x̂ is a real Gaussian and A Hermitian, and also in a multiplicative setting AUBU where A,B are fixed unitary matrices with B a multiplicative rank-1 deviation from unity, and U is a Haar distributed unitary matrix. Specifically, in each case there is a dual eigenvalue problem giving rise to a PDF of almost identical structure.

Keywords
Horn problem, Harish-Chandra Itzykson–Zuber integral
Mathematical Subject Classification 2010
Primary: 15A18, 15B52
Milestones
Received: 14 June 2019
Revised: 10 July 2019
Accepted: 1 August 2019
Published: 20 May 2020
Authors
Peter J. Forrester
ARC Centre of Excellence for Mathematical and Statistical Frontiers
Department of Mathematics and Statistics
The University of Melbourne
Australia
Jiyuan Zhang
ARC Centre of Excellence for Mathematical and Statistical Frontiers
Department of Mathematics and Statistics
The University of Melbourne
Australia