Spectral theory of the G-symmetric tridiagonal matrices related to Stahl’s counterexample

The Spectral Decomposition of Some Tridiagonal Matrices

Some properties of near-Toeplitz tridiagonal matrices with specific perturbations in the first and last main diagonal entries are considered. Applying the relation between the determinant and Chebyshev polynomial of the second kind, we first give the explicit expressions of determinant and characteristic polynomial, then eigenvalues are shown by finding the roots of the characteristic polynomia...

Fine Spectra of Tridiagonal Symmetric Matrices

Eigenvalues of symmetric tridiagonal interval matrices revisited

In this short note, we present a novel method for computing exact lower and upper bounds of a symmetric tridiagonal interval matrix. Compared to the known methods, our approach is fast, simple to present and to implement, and avoids any assumptions Our construction explicitly yields those matrices for which particular lower and upper bounds are attained.

Exponentials of Symmetric Matrices through Tridiagonal Reductions

A simple and efficient numerical algorithm for computing the exponential of a symmetric matrix is developed in this paper. For an n× n matrix, the required number of operations is around 10/3 n. It is based on the orthogonal reduction to a tridiagonal form and the Chebyshev uniform approximation of e−x on [0,∞).

Inner deflation for symmetric tridiagonal matrices

Suppose that one knows an accurate approximation to an eigenvalue of a real symmetric tridiagonal matrix. A variant of deflation by the Givens rotations is proposed in order to split off the approximated eigenvalue. Such a deflation can be used instead of inverse iteration to compute the corresponding eigenvector. © 2002 Elsevier Science Inc. All rights reserved.