Banded Hermitian Matrices, Matrix Orthogonal Polynomials, and the Toda Lattice
Spectral Theory
2026-04-14 v1 Mathematical Physics
math.MP
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.
Cite
@article{arxiv.2604.09883,
title = {Banded Hermitian Matrices, Matrix Orthogonal Polynomials, and the Toda Lattice},
author = {Charbel Abi Younes and Thomas Trogdon},
journal= {arXiv preprint arXiv:2604.09883},
year = {2026}
}