Let be a normalised standard complex Gaussian vector, and project an Hermitian matrix onto the hyperplane orthogonal to . 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 , 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 is a real Gaussian and Hermitian, and also in a multiplicative setting where are fixed unitary matrices with a multiplicative rank-1 deviation from unity, and 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.
"Corank-1 projections and the randomised Horn problem." Tunisian J. Math. 3 (1) 55 - 73, 2020. https://doi.org/10.2140/tunis.2021.3.55