In the current work, the authors present a symbolic algorithm for finding the inverse of any general nonsingular tridiagonal matrix. The algorithm is mainly based on the work presented in [Y. Huang, W.F. McColl, Analytic inversion of general tridiagonal matrices, J. Phys. A 30 (1997) 7919–7933] and [M.E.A. El-Mikkawy, A fast algorithm for evaluating nth order tridiagonal determinants, J. Comput...

[15] D. O'Leary and G. W. Stewart. Computing the eigenvalues and eigenvectors of symmetric arrowhead matrices. [17] A. Sameh and D. Kuck. A parallel QR algorithm for symmetric tridiagonal matrices. [21] Zhonggang Zeng. The acyclic eigenproblem can be reduced to the arrowhead one. [22] Hongyuan Zha. A two-way chasing scheme for reducing a symmetric arrowhead matrix to tridiagonal form. Scientic ...

In this paper, the concept of generalized spectral function is introduced for finite-order tridiagonal symmetric matrices (Jacobi matrices) with complex entries. The structure of the generalized spectral function is described in terms of spectral data consisting of the eigenvalues and normalizing numbers of the matrix. The inverse problems from generalized spectral function as well as from spec...

Tridiagonal parametrizations of linear state-space models are proposed for multivariable system identiication. The parametrizations are surjective, i.e. all systems up to a given order can be described. The parametrization is based on the fact that any real square matrix is similar to a real tridiagonal form as well as a compact tridi-agonal form. These parametrizations has signiicantly fewer p...

The Sturm sequence computation is used by the bisection method to compute eigenvalues of real symmetric tridiagonal matrices. Let Tn be a symmetric tridiagonal matrix with the diagonal elements α1, α2, . . . , αn and the off-diagonal elements β1, β2, . . . , βn−1. Given a number λ, the sequence of characteristic polynomials pj(λ) for the leading j × j principal submatrices of Tn can be computed...

This paper presents an O(n2 log n) algorithm for computing the symmetric singular value decomposition of square Hankel matrices of order n, in contrast with existing O(n3) SVD algorithms. The algorithm consists of two stages: first, a complex square Hankel matrix is reduced to a complex symmetric tridiagonal matrix using the block Lanczos method in O(n2 log n) flops; second, the singular values...

Let V denote a vector space with finite positive dimension. We consider a pair of linear transformations A : V → V and A : V → V that satisfy (i) and (ii) below: (i) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A is diagonal. (ii) There exists a basis for V with respect to which the matrix representing A is irr...

In this study, we define a n × n tridiagonal matrix which have elements of (s, t)-Pell numbers and then investigate the determinantal properties.

A parallel algorithm for solving a series of matrix equations with a constant tridiagonal matrix and different right-hand sides is proposed and studied. The process of solving the problem is represented in two steps. The first preliminary step is fixing some rows of the inverse matrix of SLAEs. The second step consists in calculating solutions for all right-hand sides. For reducing the communic...

We investigate the properties of block matrices with block banded inverses to derive efficient matrix inversion algorithms for such matrices. In particular, we derive the following: (1) a recursive algorithm to invert a full matrix whose inverse is structured as a block tridiagonal matrix; (2) a recursive algorithm to compute the inverse of a structured block tridiagonal matrix. These algorithm...