Alex Little

Products of random matrices and their real eigenvalues

Recently there has been a wave of research into products of real asymmetric random matrices. Because these random matrices are asymmetric, they have both real and complex eigenvalues, with the number of each being random. The real eigenvalues of an asymmetric random matrix interact in an interesting way with taking products, in particular, longer products tend to lead to more real eigenvalues. We look at a particular model, products of so-called "truncated orthogonal" matrices and prove a conjecture about the number of real eigenvalues and their distribution along the real line. Proving this conjecture amounted to a problem in asymptotic analysis, and I will review the tricks we used to carry this out. This was joint work with Francesco Mezzadri (Bristol) and Nicholas Simm (Sussex).
Our paper can be found here: https://arxiv.org/abs/2102.08842