Orthogonal matrix polynomials and applications

Orthogonal matrix polynomials, on the real line or on the unit circle, have properties which are natural generalizations of properties of scalar orthogonal polynomials, appropriately modified for the matrix calculus. We show that orthogonal matrix polynomials, both on the real line and on the unit circle, appear at various places and we describe some of them. The spectral theory of doubly infinite Jacobi matrices can be described using orthogonal 2×2 matrix polynomials on the real line. Scalar orthogonal polynomials with a Sobolev inner product containing a finite number of derivatives can be studied using matrix orthogonal polynomials on the real line. Orthogonal matrix polynomials on the unit circle are related to unitary block Hessenberg matrices and are very useful in multivariate time series analysis and multichannel signal processing. Finally we show how orthogonal matrix polynomials can be used for Gaussian quadrature of matrix-valued functions. Parallel algorithms for this purpose have been implemented (using PVM) and some examples are worked out.