Matrix Bispectrality of Full Rank One Algebras

TL;DR

This paper investigates the algebraic structure of full rank 1 matrix algebras and develops a method to determine their bispectrality, with applications to scalar and matrix-valued eigenvalues.

Contribution

It introduces a general framework for analyzing full rank 1 algebras and provides a novel method to verify bispectrality in matrix polynomial sub-algebras.

Findings

Developed a verification method for bispectrality in full rank 1 algebras.

Illustrated the method with examples involving scalar and matrix-valued eigenvalues.

Applied Pierce decomposition to analyze algebraic properties.

Abstract

We study algebraic properties of full rank 1 algebras in a general framework and derive a method to verify if one such matrix polynomial sub-algebra is bispectral. We give two examples illustrating the method. In the first one, we consider the eigenvalue to be scalar-valued, whereas, in the second one, we assume it to be matrix-valued. In the former example, we put forth a Pierce decomposition of that algebra.