The eigenvalues and eigenvectors of tridiagonal Toeplitz matrices are known in closed form. Explicit expressions for the structured distance to the closest normal matrix, the departure from normality, and the ϵ‐pseudospectrum are derived. Keywords: Tridiagonalmatrices, eigenvalues, recurrencerelations, Chebyshevpoly-nomials. the inverse matrices of the tridiagonal Toeplitz matrix with opposite-bordered rows are presented. In this paper, we consider an inverse problem with the k-tridiagonal Toeplitz matrices. 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. Theorem 1. =un =0, which is contrary to the deﬁnition of an eigenvector. 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. tridiagonal matrices suggested by William Trench. 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. B transformation matrices, we give the spectral decomposition of this kind of tridiagonal matrices. 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. 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. A theoretical result is obtained that under certain assumptions the explicit inverse of a k-tridiagonal Toeplitz matrix can be derived immediately. conditions for the inverse matrix to be numerically banded and may also aid in the design of preconditioners and fast algorithms. This property is in the first part of the paper used to investigate the sensitivity of the spectrum. In Section 4, the eigenvalues and eigenvectors of the tridiagonal Toeplitz matrix with opposite-bordered rows are introduced. Two numerical experiments are given to show the AMS Subject Classification (2000): 15A18, 65F15, 15A09, 15A47, 65F10. tridiagonal Toeplitz matrix with perturbed corners. Furthermore, the inverse (if the matrix is invertible), powers and a square root are also determined. Explicit inverse of a tridiagonal (p;r){Toeplitz matrix A.M. Encinas, M.J. Jim enez Departament de Matemtiques Universitat Politcnica de Catalunya Abstract Tridiagonal matrices appears in many contexts in pure and applied mathematics, so the study of the inverse … Besides, we make some analysis of these algorithms to illustrate our theoretical results. Two numerical examples are given to demonstrate the validity of our results. Keywords: matrix inversion algorithms, matrix Möbius transformations, block tridiagonal matrices, block Toeplitz matrices And also the algorithm is presented for main theorem. 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. Finally, we present numerical examples of these matrix types. 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