Determinant formulas for finite Toeplitz plus Hankel matrices with rational symbols

TL;DR

This paper extends Day's 1975 determinant formula from Toeplitz matrices to Toeplitz plus Hankel matrices with rational symbols, providing explicit formulas and insights into eigenvalue limits.

Contribution

It introduces an exact determinant formula for finite Toeplitz plus Hankel matrices with rational symbols, generalizing previous results and enabling eigenvalue analysis.

Findings

Derived an explicit determinant formula involving Fredholm determinants.

Applied the formula to analyze limiting eigenvalues.

Extended Day's classical results to a broader class of structured matrices.

Abstract

In 1975 K. Michael Day produced an exact formula for the determinants of finite Toeplitz matrices whose symbols are rational. The answer is a sum that involves powers of the roots of the numerator of the symbol and whose coefficients depend on both the roots of the numerator and denominator. In this paper we prove an analogue of Day's formula for determinants of finite Toeplitz plus Hankel matrices with rational symbols. The key to the proof is an exact formula for the finite determinants that involves a Fredholm determinant that can be explicitly computed. We apply the formula to find information about the limiting eigenvalues of the finite Toeplitz plus Hankel matrices.