FAST TRACK COMMUNICATION: Integral formulae for the eigenvalue density of complex random matrices

We show that the density of complex eigenvalues for unitary invariant ensembles of complex matrices A can be written as an integral over the eigenvalues g_j of AA†. For the standard random matrix ensembles with matrix density of the form ∏_jw(g_j), this integral can be further reduced to a twofold integral involving the Christoffel-Darboux kernel for the orthogonal polynomials associated with weight w.