Spectral Radii of Products of Random Rectangular Matrices

TL;DR

This paper investigates the asymptotic behavior of the spectral radius of products of independent random rectangular matrices with Gaussian entries, providing a comprehensive description of its limiting distribution as matrix dimensions grow.

Contribution

It extends previous results on spectral radii from square matrices to rectangular matrices, analyzing the effects of the number of matrices on the spectral radius distribution.

Findings

Derived the limiting distribution of spectral radii for products of rectangular matrices.

Showed the results generalize known square matrix cases.

Provided conditions under which the spectral radius converges or diverges.

Abstract

We consider m independent random rectangular matrices whose entries are independent and identically distributed standard complex Gaussian random variables. Assume the product of the m rectangular matrices is an n by n square matrix. The maximum absolute values of the n eigenvalues of the product matrix is called spectral radius. In this paper, we study the limiting spectral radii of the product when m changes with n and can even diverge. We give a complete description for the limiting distribution of the spectral radius. Our results reduce to those in Jiang and Qi [26] when the rectangular matrices are square ones.