On the eigenvalues of Toeplitz matrices with two off-diagonals

Abstract

Consider the Toeplitz matrix $T_n(f)$ generated by the symbol $f(\theta )={\hat{f}}_r{e}^{\mathbf{i}r\theta }+{\hat{f}}_0+{\hat{f}}_{-s}{e}^{-\mathbf{i}s\theta }$, where ${\hat{f}}_r,{\hat{f}}_0,{\hat{f}}_{-s}\in \mathbb{C}$ and $0<r<n,0<s<n$. For $r=s=1$ we have the classical tridiagonal Toeplitz matrices, for which the eigenvalues and eigenvectors are known. Similarly, the eigendecompositions are known for $1<r=s$, when the generated matrices are ``symmetrically sparse tridiagonal''. In the current paper we study the eigenvalues of $T_n(f)$ for $1\le r<s$, which are ``non-symmetrically sparse tridiagonal''. We propose an algorithm which constructs one or two ad hoc matrices smaller than $T_n(f)$, whose eigenvalues are sufficient for determining the full spectrum of $T_n(f)$. The algorithm is explained through use of a conjecture for which examples and numerical experiments are reported for supporting it and for clarifying the presentation. Open problems are briefly discussed.