Co-rank 1 projections and the randomised Horn problem

Abstract

Let x be a normalised standard complex Gaussian vector, and project an Hermitian matrix A onto the hyperplane orthogonal to 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 + bxx † , and asks for an explanation. We provide one by way of a common derivation involving the secular equations and associated Jacobians. This applies too in related setting, for example when x 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 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.

Hard Edge Statistics of Products of Pólya Ensembles and Shifted GUE's

“…The integrability for a → ∞ is given by ω (n) . Applying the definition (17) and (20), the kernel for…”

Derivative principles for invariant ensembles

“…Now for such eigenvalues to be fixed, i.e. the eigenvalue distributions are Dirac deltas, Horn's problem generalises to a sum of randomized adjoint orbits of O(n) and U(n), see [13,34,7,11]. We would like to draw also attention to some recent works discussing a multiplicative version of Horn's problem [11,33].…”

“…the eigenvalue distributions are Dirac deltas, Horn's problem generalises to a sum of randomized adjoint orbits of O(n) and U(n), see [13,34,7,11]. We would like to draw also attention to some recent works discussing a multiplicative version of Horn's problem [11,33]. For general eigenvalues, a general unitarily invariant ensemble, the Pólya ensemble, on Herm(n) have been identified in [28,12].…”