Inverse eigenvalue problems of tridiagonal symmetric matrices and tridiagonal bisymmetric matrices

Inverse Tridiagonal Z – Matrices

In this paper, we consider matrices whose inverses are tridiagonal Z–matrices. Based on a characterization of symmetric tridiagonal matrices by Gantmacher and Krein, we show that a matrix is the inverse of a tridiagonal Z–matrix if and only if, up to a positive scaling of the rows, it is the Hadamard product of a so called weak type D matrix and a flipped weak type D matrix whose parameters sat...

An inverse eigenvalue problem for symmetrical tridiagonal matrices

We consider the following inverse eigenvalue problem: to construct a symmetrical tridiagonal matrix from the minimal and maximal eigenvalues of all its leading principal submatrices. We give a necessary and sufficient condition for the existence of such a matrix and for the existence of a nonnegative symmetrical tridiagonal matrix. Our results are constructive, in the sense that they generate a...

A Scalable Eigenvalue Solver for Symmetric Tridiagonal Matrices

This paper describes a parallel eigenvalue solver for symmetric tridiagonal matrices, as implemented on an nCUBE-2 hypercube multicomputer. The algorithm is based on the split-merge technique, which uses Laguerre's iteration and exploits the separation property in order to create subtasks that can be solved independently. Performance of an initial parallel implementation of the algorithm was ne...

Symmetric Tridiagonal Inverse Quadratic Eigenvalue Problems with Partial Eigendata

In this paper we concern the inverse problem of constructing the n-by-n real symmetric tridiagonal matrices C and K so that the monic quadratic pencil Q(λ) := λI + λC + K (where I is the identity matrix) possesses the given partial eigendata. We first provide the sufficient and necessary conditions for the existence of an exact solution to the inverse problem from the self-conjugate set of pres...

Fine Spectra of Tridiagonal Symmetric Matrices