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Inverses and eigenpairs of tridiagonal Toeplitz matrix with opposite-bordered rows
Yaru Fu,Xiaoyu Jiang,Zhaolin Jiang,Seong Tae Jhang
Keywords:tridiagonal Toeplitz matrix, opposite-bordered, Fibonacci number, determinant, inverse, eigenpairs
Abstract:
      In this paper, tridiagonal Toeplitz matrix (type I, type II) with opposite-bordered rows are introduced. Main attention is paid to calculate the determinants, the inverses and the eigenpairs of these matrices. Specifically, the determinants of an n×n tridiagonal Toeplitz matrix with opposite-bordered rows can be explicitly expressed by using the (n−1)th Fibonacci number, the inversion of the tridiagonal Toeplitz matrix with opposite-bordered rows can also be explicitly expressed by using the Fibonacci numbers and unknown entries from the new matrix. Besides, we give the expression of eigenvalues and eigenvectors of the tridiagonal Toeplitz matrix with opposite-bordered rows. In addition, some algorithms are presented based on these theoretical results. Numerical results show that the new algorithms have much better computing efficiency than some existing algorithms studied recently.
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