Pfaffian point process for the Gaussian real generalised eigenvalue problem

Abstract

The generalised eigenvalues for a pair of N × N matrices (X1, X2) are defined as the solutions of the equation det (X1 - λX2) = 0, or equivalently, for X2 invertible, as the eigenvalues of X2-1. We consider Gaussian real matrices X1, X2, for which the generalised eigenvalues have the rotational invariance of the half-sphere, or after a fractional linear transformation, the rotational invariance of the unit disk. In these latter variables we calculate the joint eigenvalue probability density function, the probability pN,k of finding k real eigenvalues, the densities of real and complex eigenvalues (the latter being related to an average over characteristic polynomials), and give an explicit P..