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 $1\leq r<s$, 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.

翻译：考虑由符号$f(\theta)=\hat{f}_r e^{\mathbf{i}r\theta}+\hat{f}_0+\hat{f}_{-s} e^{-\mathbf{i}s\theta}$生成的Toeplitz矩阵$T_n(f)$，其中$\hat{f}_r, \hat{f}_0, \hat{f}_{-s} \in \mathbb{C}$且$0<r<n,~0<s<n$。当$r=s=1$时，我们得到经典的三对角Toeplitz矩阵，其特征值与特征向量已知。类似地，当$1<r=s$时，生成的矩阵为“对称稀疏三对角”形式，其特征分解亦已知。本文研究$1\leq r<s$情形下$T_n(f)$的特征值，此时矩阵呈“非对称稀疏三对角”结构。我们提出一种算法，构造一个或两个小于$T_n(f)$的特定矩阵，这些矩阵的特征值足以确定$T_n(f)$的完整谱。该算法基于一个猜想进行阐述，并通过实例与数值实验加以支持，同时澄清陈述内容。最后简要讨论若干未解决问题。