Correlation functions between singular values and eigenvalues

Abstract

Exploiting the explicit bijection between the density of singular values and the density of eigenvalues for bi-unitarily invariant complex random matrix ensembles of finite matrix size we aim at finding the induced probability measure on j eigenvalues and k singular values that we coin j,k-point correlation measure. We fully derive all j,k-point correlation measures in the simplest cases for matrices of size n = 1 and n = 2 . For n > 2 , we find a general formula for the 1, 1-point correlation measure. This formula reduces drastically when assuming the singular values are drawn from a polynomial ensemble, yielding an explicit formula in terms of the kernel corresponding to the singular value statistics. These expressions simplify even further when the singular values are drawn from a Pólya ensemble and extend known results between the eigenvalue and singular value statistics of the corresponding bi-unitarily invariant ensemble. MSC Classification: 60B20 , 15B52 , 43A90 , 42B10 , 42C05