Eigenvalues and singular values of products of rectangular Gaussian random matrices

Abstract

We derive exact analytic expressions for the distributions of eigenvalues and singular values for the product of an arbitrary number of independent rectangular gaussian random matrices in the limit of large matrix dimensions. We show that they both have power-law behavior at zero and determine the corresponding powers. We also propose a heuristic form of finite size corrections to these expressions which very well approximates the distributions for matrices of finite dimensions.

Singular Values of Products of Ginibre Random Matrices, Multiple Orthogonal Polynomials and Hard Edge Scaling Limits

“…With tools from free probability and diagrammatic expansions, one may find the limiting global eigenvalue distributions as in [8,15,16,33]. It turns out that, as in the theory of a single random matrix, the various limits exhibit a rich and interesting mathematical structure, which also show a large degree of universality, see e.g.…”

“…It is known that the limiting mean distribution of the squared singular values for the products of M Ginibre matrices blows up with a rate x −M/(M +1) near the origin (see [16,33]). Extending the notion of universality at the hard edge, we are led to the expectation that the kernels described in Theorem 5.3 should appear in more general situations of the products of independent complex random matrices, and possibly in other models of random matrix theory.…”

Products of Independent non-Hermitian Random Matrices

“…entries. See also [1], [2], [9], [5], [7], and [24] for some other results on the spectral properties of products of random matrices.…”

Asymmetric correlation matrices: an analysis of financial data

“…Also, quite importantly for our present discussion, the aforementioned equation can be solved exactly for the product of two matrices, as in equation (7). More precisely, assuming all matrix entries in both R (1) 0 and R (2) τ to be independent and identically distributed Gaussian random numbers with zero mean and unit variance, the average eigenvalue density (in the complex plane) for the k (12) matrix can be shown [28] to be:…”

“…Not many results are known on the spectra of products of random matrices (see for example [24,25,26,27]) as the one in equation (7), and most of them only describe "microscopic" spectral properties. However, in [28] an equation for the average eigenvalue density for a product of an arbitrary number of large Gaussian random matrices was derived. Such equation was derived by means of a planar diagram expansion (see [29] for a step by step introduction to this technique) under the assumption of all matrix dimensions going to infinity with their ratios kept fixed.…”