Corank-1 projections and the randomised horn problem

Abstract

Let ˆx be a normalised standard complex Gaussian vector, and project an Hermitian matrix A onto the hyperplane orthogonal to ˆx. 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 + b ˆx ˆx†, 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 ˆx is a real Gaussian and A Hermitian, and also in a multiplicative setting AU BU† where A, B are fixed unitary matrices with B a multiplicative rank-1 deviation from uni..