conditions for the inverse matrix to be numerically banded and may also aid in the design of preconditioners and fast algorithms. Besides, we make some analysis of these algorithms to illustrate our theoretical results. In this section, we give two algorithms for finding the determinant and inverse of a periodic tridiagonal Toeplitz matrix with perturbed corners of type I, which is called A. The method used is generalizable to other problems. Keywords: matrix inversion algorithms, matrix Möbius transformations, block tridiagonal matrices, block Toeplitz matrices We consider the inversion of block tridiagonal, block Toeplitz matrices and comment on the behaviour of these inverses as one moves away from the diagonal. In Section 4, the eigenvalues and eigenvectors of the tridiagonal Toeplitz matrix with opposite-bordered rows are introduced. Two numerical examples are given to demonstrate the validity of our results. Keywords: Tridiagonalmatrices, eigenvalues, recurrencerelations, Chebyshevpoly-nomials. Two numerical experiments are given to show the =un =0, which is contrary to the deﬁnition of an eigenvector. tridiagonal matrices suggested by William Trench. The eigenvalues and eigenvectors of tridiagonal Toeplitz matrices are known in closed form. In this paper, we consider an inverse problem with the k-tridiagonal Toeplitz matrices. Explicit expressions for the structured distance to the closest normal matrix, the departure from normality, and the ϵ‐pseudospectrum are derived. Main effort is made to work out those for periodic tridiagonal Toeplitz matrix with perturbed corners of type 1, since the results for type 2 matrices would follow immediately. tridiagonal linear systems in terms of Chebyshev polynomial of the third kind or the fourth kindy constructing the inverse of the. Theorem 1. tridiagonal Toeplitz matrix with perturbed corners. A theoretical result is obtained that under certain assumptions the explicit inverse of a k-tridiagonal Toeplitz matrix can be derived immediately. Furthermore, the inverse (if the matrix is invertible), powers and a square root are also determined. In recent years the invertibility of nonsingular tridiagonal or block tridiagonal matrices has been quite investigated in different ﬁelds of applied linear algebra (for historicalnotessee [8]).Several numericalmethods,moreor less efﬁcient,have risen in order to give expressions of the entries of the inverse of this kind of matrices. As matrix A is a special tridiagonal Toeplitz matrix with two perturbed corners, its eigenvalues and eigenvectors can be determined exactly [74], leading to Eq. the inverse matrices of the tridiagonal Toeplitz matrix with opposite-bordered rows are presented. We consider the inversion of block tridiagonal, block Toeplitz matrices and comment on the behaviour of these inverses as one moves away from the diagonal. Using matrix Mobius transformations, we first present an representation (with respect to the number of block rows and block columns) for the inverse matrix and subsequently use this representation to characterize the inverse matrix. B transformation matrices, we give the spectral decomposition of this kind of tridiagonal matrices. This property is in the first part of the paper used to investigate the sensitivity of the spectrum. Firstly, based on Theorem 1, we give an algorithm for computing determinant of A: AMS Subject Classification (2000): 15A18, 65F15, 15A09, 15A47, 65F10. Finally, we present numerical examples of these matrix types. And also the algorithm is presented for main theorem. Explicit inverse of a tridiagonal (p;r){Toeplitz matrix A.M. Encinas, M.J. 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