Spectral distributions and isospectral sets of tridiagonal matrices

TL;DR

This paper explores the relationship between probability distributions and tridiagonal matrices, providing a new topological framework and applying it to analyze isospectral sets with implications in inverse problems and integrable systems.

Contribution

It offers an intrinsic topological description of sequences of distributions related to tridiagonal matrices and links these to the Euler characteristic of isospectral sets.

Findings

Topological structure of distribution sequences is characterized.

Euler characteristic of generic isospectral sets equals a tangent number.

Analytical tools developed for studying ensembles of tridiagonal matrices.

Abstract

We analyze the correspondence between finite sequences of finitely supported probability distributions and finite-dimensional, real, symmetric, tridiagonal matrices. In particular, we give an intrinsic description of the topology induced on sequences of distributions by the usual Euclidean structure on matrices. Our results provide an analytical tool with which to study ensembles of tridiagonal matrices, important in certain inverse problems and integrable systems. As an application, we prove that the Euler characteristic of any generic isospectral set of symmetric, tridiagonal matrices is a tangent number.