Diagonal symmetrisation of tridiagonal Toeplitz matrices

TL;DR

This paper presents a comprehensive framework for real tridiagonal Toeplitz matrices in the symmetrizable regime, providing explicit eigenvalues, determinants, and inverse formulas, with applications to extremal eigenvalues and special cases like repunit matrices.

Contribution

It introduces explicit eigenpairs, determinant formulas, and inverse expressions for symmetrizable tridiagonal Toeplitz matrices, including applications to extremal eigenvalues and repunit matrices.

Findings

Explicit eigenpairs and determinants derived

Closed-form inverse formulas provided

Special case analysis for repunit matrices

Abstract

We develop a self-contained framework for real tridiagonal Toeplitz matrices $A_n(a,b,c)$ (diagonal $b$, subdiagonal $a$, superdiagonal $c$) in the symmetrisable regime $ac>0$. A diagonal similarity transforms $A_n(a,b,c)$ into a symmetric Toeplitz matrix, yielding explicit eigenpairs, a Chebyshev determinant/characteristic polynomial formula, and a closed Green kernel for the inverse. As an application we give sharp extremal eigenvalue and conditioning formulae in the natural weighted Hilbert space induced by this similarity. Specialising to the classical repunit matrix $A_n(d,d+1,1)$, we show that $\det(A_n(d,d+1,1))=1+d+\cdots+d^{n}$ and obtain a finite cosine product factorisation of this repunit polynomial, together with quantitative bounds and an explicit inverse in terms of repunits.