Probability that product of real random matrices have all eigenvalues real tend to 1

TL;DR

This paper investigates the probability that the product of i.i.d. real random matrices has all real eigenvalues, proving the conjecture that this probability tends to 1 when the distribution has an atom.

Contribution

It proves the conjecture that the probability approaches 1 for products of real random matrices with an atomic distribution.

Findings

Probability tends to 1 when the distribution has an atom.

Supports the conjecture for fixed-size real matrices.

Provides conditions under which all eigenvalues are real asymptotically.

Abstract

In this article we consider products of real random matrices with fixed size. Let $A_1,A_2, \dots $ be i.i.d $k \times k$ real matrices, whose entries are independent and identically distributed from probability measure $\mu$. Let $X_n = A_1A_2\dots A_n$. Then it is conjectured that $$\mathbb{P}(X_n \text{ has all real eigenvalues}) \rightarrow 1 \text{ as } n \rightarrow \infty.$$ We show that the conjecture is true when $\mu$ has an atom.