Statistics of the non-zero eigenvalues and singular values of low-rank random matrices with non-negative entries

TL;DR

This paper analytically derives the probability distributions and moments of eigenvalues and singular values of low-rank non-negative random matrices with fixed row sums, applicable in various fields like Markov chains and social networks.

Contribution

It provides the first finite-size analytical formulas for the eigenvalue and singular value statistics of such matrices, using Dirichlet distributions.

Findings

Analytical formulas match numerical simulations accurately.

Results applicable to matrices with fixed rank and prescribed row sums.

Provides insights into spectral properties relevant for applications in economics and social networks.

Abstract

We compute analytically the probability distribution and moments of the sum and product of the non-zero eigenvalues and singular values of random matrices with (i) non-negative entries, (ii) fixed rank, and (iii) prescribed sums of the entries in each row. Applications of such matrices are discussed in the context of Markov chains, economics and social networks to name a few. All results are valid at finite matrix size and are given in terms of the statistics of vectors of general Dirichlet random variables. Analytical results are corroborated by numerical simulations throughout with excellent agreement.