Diagonal symmetrisation of tridiagonal Toeplitz 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.