Noncirculant Toeplitz matrices all of whose powers are Toeplitz

Let a, b and c be fixed complex numbers. Let M _n (a, b, c) be the n × n Toeplitz matrix all of whose entries above the diagonal are a, all of whose entries below the diagonal are b, and all of whose entries on the diagonal are c. For 1 ⩽ k ⩽ n, each k × k principal minor of M _n (a, b, c) has the same value. We find explicit and recursive formulae for the principal minors and the characteristic polynomial of M _n (a, b, c). We also show that all complex polynomials in M _n (a, b, c) are Toeplitz matrices. In particular, the inverse of M _n (a, b, c) is a Toeplitz matrix when it exists.     

A note on inversion of Toeplitz matrices

It is shown that the invertibility of a Toeplitz matrix can be determined through the solvability of two standard equations. The inverse matrix can be denoted as a sum of products of circulant matrices and upper triangular Toeplitz matrices. The stability of the inversion formula for a Toeplitz matrix is also considered.     

Analytical inversion of general periodic tridiagonal matrices

In this paper we present an analytical forms for the inversion of general periodic tridiagonal matrices, and provide some very simple analytical forms which immediately lead to closed formulae for some special cases such as symmetric or perturbed Toeplitz for both periodic and non-periodic tridiagonal matrices. An efficient computational algorithm for finding the inverse of any general periodic tridiagonal matrices from the analytical form is given, it is suited for implementation using Computer Algebra systems such as MAPLE, MATLAB, MACSYMA, and MATHEMATICA. An example is also given to illustrate the algorithm.     

Fast inversion algorithms of Toeplitz-plus-Hankel matrices

Summary The paper deals with the problems of fast inversion of matricesA=T+H, whereT is Toeplitz andH is Hankel. Several algorithms are presented and compared, among them algorithms working for arbitrary strongly nonsingular matricesA=T+H.     

三对角与五对角Toeplitz矩阵求逆的算法

提出了一种求三对角与五对角Toeplitz矩阵逆的快速算法,其思想为先将Toeplitz矩阵扩展为循环矩阵,再快速求循环矩阵的逆,进而运用恰当矩阵分块求原Toeplitz矩阵的逆的算法.算法稳定性较好且复杂度较低.数值例子显示了算法的有效性和稳定性,并指出了算法的适用范围.     

Eigenvector sensitivity under general and structured perturbations of tridiagonal Toeplitz‐type matrices

The sensitivity of eigenvalues of structured matrices under general or structured perturbations of the matrix entries has been thoroughly studied in the literature. Error bounds are available, and the pseudospectrum can be computed to gain insight. Few investigations have focused on analyzing the sensitivity of eigenvectors under general or structured perturbations. This paper discusses this sensitivity for tridiagonal Toeplitz and Toeplitz‐type matrices.     

Eigenvalues of Hermitian Toeplitz matrices with smooth simple-loop symbols

The paper presents higher-order asymptotic formulas for the eigenvalues of large Hermitian Toeplitz matrices with moderately smooth symbols which trace out a simple loop on the real line. The formulas are established not only for the extreme eigenvalues, but also for the inner eigenvalues. The results extend and make more precise existing results, which so far pertain to banded matrices or to matrices with infinitely differentiable symbols. Also given is a fixed-point equation for the eigenvalues which may be solved numerically by an iteration method.     

Max-algebra and pairwise comparison matrices, II

This paper is a continuation of our 2004 paper “Max-algebra and pairwise comparison matrices”, in which the max-eigenvector of a symmetrically reciprocal matrix was used to approximate such a matrix by a transitive matrix. This approximation was based on minimizing the maximal relative error. In a later paper by Dahl a different error measure was used and led to a slightly different approximating transitive matrix. Here some geometric properties of this approximation problem are discussed. These lead, among other results, to a new characterization of a max-eigenvector of an irreducible nonnegative matrix. The case of Toeplitz matrices is discussed in detail, and an application to music theory that uses Toeplitz symmetrically reciprocal matrices is given.     

Ovals of Cassini for Toeplitz matrices

Both the Gershgorin and Brauer eigenvalue inclusion sets reduce to a single disc when applied to a matrix with a constant main diagonal. We show that a union of ovals of Cassini containing all the eigenvalues of such a matrix can nonetheless be derived and that this union is guaranteed to lie inside the Brauer set. We then apply our results to Toeplitz matrices.     

On the asymptotic spectrum of Hermitian block Toeplitz matrices with Toeplitz blocks

We study the asymptotic behaviour of the eigenvalues of Hermitian block Toeplitz matrices , with Toeplitz blocks. Such matrices are generated by the Fourier coefficients of an integrable bivariate function , and we study their eigenvalues for large and , relating their behaviour to some properties of as a function; in particular we show that, for any fixed , the first eigenvalues of tend to , while the last tend to , so extending to the block case a well-known result due to Szegö. In the case the 's are positive-definite, we study the asymptotic spectrum of , where is a block Toeplitz preconditioner for the conjugate gradient method, applied to solve the system , obtaining strict estimates, when and are fixed, and exact limit values, when and tend to infinity, for both the condition number and the conjugate gradient convergence factor of the previous matrices. Extensions to the case of a deeper nesting level of the block structure are also discussed.

     

A symmetry exploiting Lanczos method for symmetric Toeplitz matrices

Several methods for computing the smallest eigenvalues of a symmetric and positive definite Toeplitz matrix T have been studied in the literature. Most of them share the disadvantage that they do not reflect symmetry properties of the corresponding eigenvector. In this note we present a Lanczos method which approximates simultaneously the odd and the even spectrum of T at the same cost as the classical Lanczos approach.     

Eigenvalues and eigenvectors of banded Toeplitz matrices and the related symbols

It is known that for a tridiagonal Toeplitz matrix, having on the main diagonal the constant a₀ and on the two first off‐diagonals the constants a₁(lower) and a₋₁(upper), which are all complex values, there exist closed form formulas, giving the eigenvalues of the matrix and a set of associated eigenvectors. For example, for the 1D discrete Laplacian, this triple is (a₀,a₁,a₋₁)=(2,−1,−1). In the first part of this article, we consider a tridiagonal Toeplitz matrix of the same form (a₀,a_ω,a_−ω), but where the two off‐diagonals are positioned ω steps from the main diagonal instead of only one. We show that its eigenvalues and eigenvectors can also be identified in closed form and that interesting connections with the standard Toeplitz symbol are identified. Furthermore, as numerical evidences clearly suggest, it turns out that the eigenvalue behavior of a general banded symmetric Toeplitz matrix with real entries can be described qualitatively in terms of the symmetrically sparse tridiagonal case with real a₀, a_ω=a_−ω, ω=2,3,…, and also quantitatively in terms of those having monotone symbols. A discussion on the use of such results and on possible extensions complements the paper.     

A note on the spectral distribution of toeplitz matrices

An elementary and direct proof of the Szegö formula is given, for both eigen and singular values. This proof, which is based on tools from linear algebra and does not rely on the theory of Fourier series, simultaneously embraces multilevel Toeplitz matrices, block Toeplitz matrices and combinations of them. The assumptions on the generating

function f are as weak as possible; indeedf is a matrix-valued function of p variables, and it is only supposed to be integrable. In the case of singular values f(x), and hence the block p-level Toeplitz matrices it generates, are not even supposed to be square matrices. Moreover, in the asymptotic formulas for eigen and singular values the test functions involved are not required to have compact support.     

Modified SMS method for computing outer inverses of Toeplitz matrices

We introduce a new algorithm based on the successive matrix squaring (SMS) method. This algorithm uses the strategy of ε-displacement rank in order to find various outer inverses with prescribed ranges and null spaces of a square Toeplitz matrix. Using the idea of displacement theory which decreases the memory space requirements as well as the computational cost, our method tends to be very effective for Toeplitz matrices.