  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 find an expression for the $1,k$-point correlation
measure which simplifies drastically when assuming that the singular values
follow a polynomial ensemble, yielding a closed formula in terms of the kernel
corresponding to the determinantal point process of the singular value
statistics. These expressions simplify even further when the singular values
are drawn from a P\'{o}lya ensemble and extend known results between the
eigenvalue and singular value statistics of the corresponding bi-unitarily
invariant ensemble.
