Mario Kieburg: Correlations between Singular Values and Eigenvalues of Complex Matrices

Abstract:

 The study of the relation between singular values and eigenvalues of square matrices dates back to Schur who has already noticed that apart from restrictions of their domains not much can be said about it. This was later confirmed by Weyl who found a whole range of inequalities between these two set of spectral quantities. The situation changes drastically when going over from deterministic to random matrices. Less than ten years ago Holger Kösters and I found a bijective relation between the joint probabilities densities of the singular values and the eigenvalues for bi-unitarily invariant random matrices, meaning the probability density is independent of the singular vectors. Since this bijection is very explicit Matthias Allard and I could recently deduce the first explicit joint correlation functions of these two sets. I will report about this progress.