Pfaffian point process for the Gaussian real generalised eigenvalue problem

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The generalised eigenvalues for a pair of N× N matrices (X₁,X₂) are defined as the solutions of the equation (X₁-λX₂)=0, or equivalently, for X₂ invertible, as the eigenvalues of X₂⁻¹X₁. 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 Pfaffian formula for the higher correlation functions ρ₍ₖ₁,ₖ₂₎. A limit theorem for pₙ,ₖ is proved, and the scaled form of ρ₍ₖ₁,ₖ₂₎ is shown to be identical to the analogous limit for the correlations of the eigenvalues of real Gaussian matrices. We show that these correlations satisfy sum rules characteristic of the underlying two-component Coulomb gas.