Banded Hermitian Matrices, Matrix Orthogonal Polynomials, and the Toda Lattice

TL;DR

This paper develops spectral theory for finite Hermitian banded matrices using matrix orthogonal polynomials, providing reconstruction procedures, conditions for spectral measures, and exploring links to algorithms and the Toda lattice.

Contribution

It introduces explicit reconstruction methods for banded matrices from spectral data and connects spectral analysis with block algorithms and Toda lattice dynamics.

Findings

Explicit reconstruction procedure from spectral measure.

Necessary and sufficient conditions for spectral measures.

Connections established between spectral theory, algorithms, and Toda lattice.

Abstract

We study the direct and inverse spectral theory for a class of finite Hermitian banded matrices. Using the theory of matrix orthogonal polynomials, we provide an explicit procedure for reconstructing a banded matrix from a matrix-valued measure that encodes its spectral data. We establish necessary and sufficient conditions for a measure to be the spectral measure of a matrix in the examined class. We further analyze the connections between this spectral analysis, block tridiagonalization algorithms, and the Toda lattice evolution on banded matrices.