Diagonal symmetrisation of tridiagonal Toeplitz matrices
Johann Verwee
TL;DR
The paper develops a self-contained diagonal symmetrisation framework for real tridiagonal Toeplitz matrices in the ac>0 regime, showing a diagonal similarity to a symmetric Toeplitz matrix $S_n(b,s)$ with $s=\sqrt{ac}$ and establishing a weighted self-adjoint structure. This reduces spectral questions to the symmetric case and yields explicit eigenpairs, a Chebyshev-determinant formula, and a closed inverse kernel; it also provides sharp extremal eigenvalue bounds and a weighted condition number. The repunit specialization $V_n(d)=A_n(d,d+1,1)$ yields a finite cosine product for the determinant, a closed-form weighted conditioning bound, and an explicit inverse in terms of repunits, tying Chebyshev polynomials to repunit polynomials. Overall, the work offers a compact, self-contained approach with concrete spectral data and inverse formulas, plus a practically informative repunit model exhibiting rapid decay in the inverse.
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.