Equality of Lyapunov and stability exponents for products of isotropic random matrices

TL;DR

This paper proves the equality of Lyapunov and stability exponents for products of i.i.d. isotropic random matrices, derives their asymptotic spectral distributions, and explores the probability of real eigenvalues in large products.

Contribution

It establishes the equality of Lyapunov and stability exponents for isotropic matrices and analyzes their spectral properties and eigenvalue distributions.

Findings

Lyapunov and stability exponents are equal for isotropic matrices.

Asymptotic distribution of singular values and eigenvalues derived.

Probability of all real eigenvalues approaches one as product size increases.

Abstract

We show that Lyapunov exponents and stability exponents are equal in the case of product of $i.i.d$ isotropic(also known as bi-unitarily invariant) random matrices. We also derive aysmptotic distribution of singular values and eigenvalues of these product random matrices. Moreover, Lyapunov exponents are distinct, unless the random matrices are random scalar multiples of Haar unitary matrices or orthogonal matrices. As a corollary of above result, we show probability that product of $n$ $i.i.d$ real isotropic random matrices has all eigenvalues real goes to one as $n \to \infty$. Also, in the proof of a lemma, we observe that a real (complex) Ginibre matrix can be written as product of a random lower triangular matrix and an independent truncated Haar orthogonal (unitary) matrix.