Co-rank 1 projections and the randomised Horn problem

Let $\widehat{\mathit{x}}$ be a normalised standard complex Gaussian vector, and project an Hermitian matrix $A$ onto the hyperplane orthogonal to $\widehat{\mathit{x}}$. In a recent paper Faraut [Tunisian J. Math. $\mathbf{\text{1}}$ (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 \hat{\boldsymbol x} \hat{\boldsymbol x}^\dagger$, and asks for an explanation. We provide this by way of a common derivation involving the secular equations and associated Jacobians. This applies too in related setting, for example when $\widehat{\mathit{x}}$ 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.