Speaker
Description
The deep connection between Krylov methods, scalar orthogonal polynomials, and moment matrices is well established, particularly for Hermitian and unitary matrices. In this talk, we extend this framework to block Krylov methods and orthogonal matrix polynomials.
By representing the elements of a block Krylov subspace via matrix polynomials, we consider the matrix-valued inner product introduced in [1], which, under a non-degeneracy assumption, defines a linear isometry. This establishes a one-to-one correspondence between orthonormal matrix polynomials and orthonormal bases of the block Krylov subspace.
For normal matrices, the block Gauss discretization [1,2] of such an inner product admits an integral representation familiar from the theory of orthogonal matrix polynomials. As an application, we extend a Szegő-type short recurrence, originally developed for matrix polynomials on the unit circle [3], to the block Arnoldi algorithm applied to unitary matrices.
Finally, we analyze the structure of the block moment matrix and explore its connection to orthogonal matrix polynomials and recurrence coefficients via a Cholesky-like factorization.
[1] Lund, K.: A New Block Krylov Subspace Framework with Applications to Functions of Matrices Acting on Multiple Vectors, Phd thesis, Temple University and Bergische Universität Wuppertal, (2018)
[2] Zimmerling, J., Druskin, V., Simoncini, V.: Monotonicity, bounds and acceleration of block Gauss and Gauss-Radau quadrature for computing $B^T\phi(A)B$, J. Sci. Comput., 103 (2025)
[3] Sinap, A., Van Assche, W.: Orthogonal matrix polynomials and applications, J. Comput. Appl. Math. 66 (1996)