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

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 $X_n=A_1{A}_2\dots A_n$. Then it is conjectured that $$\mathbb{P}(X_n\text{ has all real eigenvalues})\to 1\text{ as }n\to \infty .$$ We show that the conjecture is true when $\mu$ has an atom.

arXiv commentBased on author's Ph.D thesis arXiv:1602.05298

References in corpus (7)

• Universal distribution of Lyapunov exponents for products of Ginibre matrices
• Probability of all eigenvalues real for products of standard Gaussian matrices
• On the number of real eigenvalues of products of random matrices and an application to quantum entanglement
• Lyapunov exponents for products of rectangular real, complex and quaternionic Ginibre matrices
• Real eigenvalues of non-Gaussian random matrices and their products
• The Probability That All Eigenvalues are Real for Products of Truncated Real Orthogonal Random Matrices
• On critical points of random polynomials and spectrum of certain products of random matrices

Cited by in corpus (3)

• The Product of $m$ real $N\times N$ Ginibre matrices: Real eigenvalues in the critical regime $m=O(N)$
• On the real spectrum of a product of Gaussian random matrices
• Real eigenvalue statistics for products of asymmetric real Gaussian matrices