The EigenProblem for Some Special Near-Toeplitz Centro-Skew Tridiagonal Matrices (original) (raw)

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.

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.

EIGENVALUES FOR TRIDIAGONAL 3-TOEPLITZ MATRICES

Journal of Mahani Mathematical Research, 2021

In this paper, we study the eigenvalues of real tridiagonal 3-Toeplitz matrices of different order. When the order of a tridiagonal 3-Toeplitz matrix is n = 3k + 2, the eigenvalues were found explicitly. Here, we consider the distribution of eigenvalues for a tridiagonal 3-Toeplitz matrix of orders n = 3k and n = 3k + 1. We explain our method by finding roots of a combination of Chebyshev polynomials of the second kind. This distribution solves the eigenproblem for integer powers of such 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.

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.

Spectrum of certain tridiagonal matrices when their dimension goes to infinity

Linear Algebra and its Applications, 2009

This paper deals with the spectra of matrices similar to infinite tridiagonal Toeplitz matrices with perturbations and with positive off-diagonal elements. We will discuss the asymptotic behavior of the spectrum of such matrices and we use them to determine the values of a matrix function, for an entire function. In particular we determine the matrix powers and matrix exponentials.

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.

A recurrence relation for singular real symmetric Toeplitz matrices

IEEE Transactions on Signal Processing, 2000

We extend a recurrence relation between determinants of real symmetric Toeplitz matrices to the case where these matrices are singular. The relation exploits the even-odd structure of Toeplitz matrices and contains two factors: one corresponding to the even and the other corresponding to the odd part of their spectrum.