#### 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 $\stackrel{̂}{x}$ be a normalised standard complex Gaussian vector, and project an Hermitian matrix $A$ onto the hyperplane orthogonal to $\stackrel{̂}{x}$. 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 $A+b\stackrel{̂}{x}{\stackrel{̂}{x}}^{†}$, 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 $\stackrel{̂}{x}$ is a real Gaussian and $A$ Hermitian, and also in a multiplicative setting $AUB{U}^{†}$ 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