Random block matrices and matrix orthogonal polynomials

TL;DR

This paper links random block matrices to matrix orthogonal polynomials, showing eigenvalues can be approximated by roots of these polynomials, and derives their asymptotic spectral distribution.

Contribution

It establishes a novel connection between random block matrices and matrix orthogonal polynomials, providing new insights into their spectral properties.

Findings

Eigenvalues approximated by roots of matrix orthogonal polynomials

Derived asymptotic spectral distribution with explicit density

Established a new relation between random matrices and matrix orthogonal polynomials

Abstract

In this paper we consider random block matrices, which generalize the general beta ensembles, which were recently investigated by Dumitriu and Edelmann (2002, 2005). We demonstrate that the eigenvalues of these random matrices can be uniformly approximated by roots of matrix orthogonal polynomials which were investigated independently from the random matrix literature. As a consequence we derive the asymptotic spectral distribution of these matrices. The limit distribution has a density, which can be represented as the trace of an integral of densities of matrix measures corresponding to the Chebyshev matrix polynomials of the first kind. Our results establish a new relation between the theory of random block matrices and the field of matrix orthogonal polynomials, which have not been explored so far in the literature.