Block Krylov subspaces and orthogonal matrix polynomials: a structural correspondence with applications to unitary matrices

TL;DR

This paper establishes a structural link between block Krylov subspaces and matrix orthogonal functions, enabling new analysis and algorithms for unitary matrices through polynomial and rational functions.

Contribution

It introduces a unified framework connecting block Krylov subspaces with matrix orthogonal polynomials, including rational and Laurent polynomials, with applications to unitary matrices.

Findings

Polynomial block Krylov subspaces are isometrically isomorphic to spaces of matrix polynomials.

The inner product for normal matrices can be represented via a spectral matrix measure.

In the unitary case, Szeg"H{o} recurrence and CMV frameworks are extended to block Krylov subspaces.

Abstract

We study the connection between block Krylov subspaces and matrix orthogonal functions. Under a no-deflation assumption, we show that polynomial block Krylov subspaces are isometrically isomorphic to spaces of matrix polynomials of bounded degree, providing a unified framework for the analysis and construction of orthonormal bases and recurrence relations. The same correspondence holds for rational block Krylov subspaces and matrix-valued rational functions, and in the extended Krylov setting this leads naturally to Laurent matrix polynomials. When the matrix $A$ is normal, we prove that the induced inner product admits a representation in terms of a discrete spectral matrix measure, extending a classical result for Hermitian matrices. In the unitary case, where the measure is supported on the unit circle, this connection allows us to transfer the Szeg\H{o} recurrence for orthogonal matrix polynomials and the CMV framework for Laurent matrix polynomials to the block Krylov setting, yielding efficient procedures for the orthogonalization of polynomial and extended block Krylov subspaces.