In this paper, for a class of perturbed Toeplitz periodic tridiagonal (PTPT) matrices, some properties, including the determinant, the inverse matrix, the eigenvalues and the eigenvectors, are studied in detail. Specifically, the determinant of the PTPT matrix can be explicitly expressed using the well-known Fibonacci numbers; the inverse of the PTPT matrix can also be explicitly expressed using the Lucas number and only four elements in the PTPT matrix. Eigenvalues and eigenvectors can be obtained under certain conditions. In addition, some algorithms are presented based on these theoretical results. Comparison of our new algorithms and some recent works is given. Numerical results confirm our new theoretical results and show that the new algorithms not only can obtain accurate results but also have much better computing efficiency than some existing algorithms studied recently.

本文针对一类摄动的Toeplitz周期三对角线（PTPT）矩阵，详细研究了一些性质，包括行列式，逆矩阵，特征值和特征向量。具体而言，可以使用众所周知的斐波纳契数明确表示PTPT矩阵的行列式；PTPT矩阵的逆还可以使用卢卡斯数和PTPT矩阵中的仅四个元素来明确表示。特征值和特征向量可以在某些条件下获得。另外，基于这些理论结果提出了一些算法。比较了我们的新算法和一些近期的工作。