Properties of a class of perturbed Toeplitz periodic tridiagonal matrices

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.