Eigenvalues and Singular Values of Products of Rectangular Gaussian Random Matrices

TL;DR

This paper derives exact formulas for the eigenvalue and singular value distributions of products of rectangular Gaussian matrices, revealing power-law behaviors and proposing finite size corrections for practical matrix dimensions.

Contribution

It provides the first exact analytic expressions for these distributions and introduces heuristic finite size corrections, advancing understanding of Gaussian matrix products.

Findings

Eigenvalue and singular value distributions exhibit power-law behavior at zero.

Derived exact formulas for large matrix dimensions.

Proposed heuristic finite size correction models.

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.