Evaluation of spectrum of 2-periodic tridiagonal-Sylvester matrix (original) (raw)

On periodic block-tridiagonal matrices

Linear Algebra and its Applications, 1992

Periodic block-tridiagonal matrices are defined, and conditions are given for factorizing their characteristic polynomial by means of the zeros of Chebyshev polynomials of the second kind. These conditions are expressed by the block centrosymmetry of certain submatrices along the main diagonal. The main theorems can be considered as generalizations of an earlier result published by Elsner and Redheffer and by Rozsa. 1. *Work supported by the Progetto Finalizzato Calcolo Parallelo e Sistemi Inforrnatici of CNR. ' Visiting Professor at the University of Pisa under the support of GNIM-CNR.

On Inverses and Eigenpairs of Periodic Tridiagonal Toeplitz Matrices with Perturbed Corners

Journal of Applied Analysis & Computation

In this paper, we derive explicit determinants, inverses and eigenpairs of periodic tridiagonal Toeplitz matrices with perturbed corners of Type I. The Mersenne numbers play an important role in these explicit formulas derived. Our main approaches include clever uses of the Schur complement and matrix decomposition with the Sherman-Morrison-Woodbury formula. Besides, the properties of Type II matrix can be also obtained, which benefits from the relation between Type I and II matrices. Lastly, we give three algorithms for these basic quantities and analyze them to illustrate our theoretical results.

Determinants and inverses of perturbed periodic tridiagonal Toeplitz matrices

Advances in Difference Equations

In this paper, we deal mainly with a class of periodic tridiagonal Toeplitz matrices with perturbed corners. By matrix decomposition with the Sherman–Morrison–Woodbury formula and constructing the corresponding displacement of matrices we derive the formulas on representation of the determinants and inverses of the periodic tridiagonal Toeplitz matrices with perturbed corners of type I in the form of products of Fermat numbers and some initial values. Furthermore, the properties of type II matrix can be also obtained, which benefits from the relation between type I and II matrices. Finally, we propose two algorithms for computing these properties and make some analysis about them to illustrate our theoretical results.

A formula for Eigenpairs of certain symmetric tridiagonal matrices

Bulletin of the Australian Mathematical Society, 1997

A closed form expression is given for the eigenvalues and eigenvectors of a symmetric tridiagonal matrix of odd order whose diagonal elements are all equal and whoes superdiagonal elements alternate between the values c and d. An implicit formula is given for the even order case.

Direct and inverse spectral problems for a class of non-self-adjoint periodic tridiagonal matrices

Linear Algebra and Its Applications, 2013

The spectral properties of a class of band matrices are investigated. The reconstruction of matrices of this special class from given spectral data is also studied. Necessary and sufficient conditions for that reconstruction are found. The obtained results extend some results on the direct and inverse spectral problems for periodic Jacobi matrices and for some non-self-adjoint tridiagonal matrices.

Spectral characterizations and integer powers of tridiagonal 2-Toeplitz matrices

In this note, we consider real nonsymmetric tridiagonal 2-Toeplitz matrices boldBn\bold{B}_nboldBn. First we give the asymptotic spectral and singular value distribution of the whole matrix-sequence boldBnn\{\bold{B}_n\}_nboldBnn, which is described via two eigenvalue functions of a 2times22\times 22times2 matrix-valued symbol. In connection with the above findings, we provide a characterization of the eigenvalues and eigenvectors of real tridiagonal 2-Toeplitz matrices boldBn\bold{B}_nboldBn of even order, thatcan be turned into a numerical effective scheme for the computation of all the entries of boldBnl\bold{B}_n^lboldBnl, nnn even and lll positive and small compared to nnn. We recall that a corresponding eigenvalue decomposition for odd order tridiagonal 2-Toeplitz matrices was found previously, while, for even orders, an implicit formula for all the eigenvalues is obtained. AMS SC 15B05; 15A18; 65F15.

Eigenvalues of 2-tridiagonal Toeplitz matrix

Journal of Applied Mathematics and Computational Mechanics, 2015

In this article an explicit formula for eigenvalues of a 2-tridiagonal Toeplitz matrix can be derived on the basis of a certain relation between the determinant of this matrix and the determinant of a pertinent tridiagonal matrix. It can be pointed out that the problem is investigated without imposing any conditions on the elements of matrix.

Eigenpairs of a family of tridiagonal matrices: three decades later

Acta Mathematica Hungarica, 2019

This survey paper summarizes the more important recent applications of the eigenpairs formulas for a family of tridiagonal matrices based on Losonczi's seminal work of almost thirty years ago, which not only seems to have been largely ignored, but has also been re-cast or rediscovered in alternative guises by various authors since. In the course of presenting these applications, we also make contact with earlier more specific applications where Losonczi's work could have been applied to yield the results more quickly. Many of the recent applications in physics and engineering cite less general work, which followed Losonczi more than a decade later.