On inversion of block Toeplitz matrices

In [1] we proved that each inverse of a Toeplitz matrix can be constructed via three of its columns, and thus, a parametrization of the set of inverses of Toeplitz matrices was obtained. A generalization of these results to block Toeplitz matrices is the main aim of this paper.     

On inverse eigenvalue problems for block Toeplitz matrices with Toeplitz blocks

We propose an algorithm for solving the inverse eigenvalue problem for real symmetric block Toeplitz matrices with symmetric Toeplitz blocks. It is based upon an algorithm which has been used before by others to solve the inverse eigenvalue problem for general real symmetric matrices and also for Toeplitz matrices. First we expose the structure of the eigenvectors of the so-called generalized centrosymmetric matrices. Then we explore the properties of the eigenvectors to derive an efficient algorithm that is able to deliver a matrix with the required structure and spectrum. We have implemented our ideas in a Matlab code. Numerical results produced with this code are included.     

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.

     

On the distribution singular values of Toeplitz matrices

We prove a second order formula concerning distribution of singular values of Toeplitz matrices in some cases when conditions of the H. Widom Theorem are not satisfied.

     

Toeplitz matrices with Toeplitz inverses revisited

A necessary and sufficient condition derived by Huang and Cline for a nonsingular Toeplitz matrix to have a Toeplitz inverse is shown to hold under more general hypotheses than indicated by them.     

On the eigenvalue problem for Toeplitz band matrices

A formula is given for the characteristic polynomial of an nth order Toeplitz band matrix, with bandwidth k < n, in terms of the zeros of a kth degree polynomial with coefficients independent of n. The complexity of the formula depends on the bandwidth k, and not on the order n. Also given is a formula for eigenvectors, in terms of the same zeros and k coefficients which can be obtained by solving a k × k homogeneous system.     

Toeplitz matrices commuting with tridiagonal matrices

we prove that if R is a nonscalar Toeplitz matrix R_{i, j}=r_?i?j? which commutes with a tridiagonal matrix with simple spectrum, then

$\text{r}{}_{\mathrm{k}}\text{r}{}_1\text{=}\text{u}{}_{\mathrm{k-1}}\text{r}{}_2\text{r}{}_1\text{cos p}\text{u}{}_{\mathrm{k-1}}\text{(cos p)}$

, k=4, 5,…, with U_k the Chebychev polynomial of the second kind, where p is determined from

$\text{cos p=}\text{1}\text{2}\text{r}{}^2{}_1\text{?r}{}_1\text{r}{}_3\text{r}{}^2{}_2\text{?r}{}_1\text{r}{}_3$

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Type I Toeplitz algebras

The class of Toeplitz algebras associated to ordered groups is important in the analysis of Toeplitz operators on the generalised Hardy spaces defined by such groups. The conditions under which these Toeplitz algebras are Type I C^*-algebras are investigated.     

On conditions for permutability of Toeplitz and Hankel matrices

The problem of describing pairs of commuting matrices (T, H), where T and H are a Toeplitz and a Hankel matrix, respectively, is examined. Several families of such pairs are indicated.     

On symmetric algebras of fuzzy sets

In this paper all symmetric algebras of fuzzy sets taking values on a distributive directly nondecomposable lattice with universal bounds 1 and 0, and their classes with respect to automorphisms, are studied. Special attention is devoted to the case where fuzzy sets take values on [0, 1].     

On unitary interpolants and Fredholm infinite block Toeplitz matrices

Conditions for the existence and uniqueness of unitary n × n matrix valued functions f on the unit circle with prescribed Fourier coefficients f_j for j 0 are given (in terms of infinite block Hankel matrices based on the prescribed coefficients f₀,f₁, ) for a natural class of functions. A unitary function belongs to this class if and only if it admits a generalized factorization (in a sense which will be made precise in the paper) or equivalently if and only if any one (and hence both) of the two Toeplitz operators defined by the function are Fredholm. In particular this class includes all continuous unitary n × n matrix valued functions. It is shown that the nonnegative factorization indices of every such unitary f are uniquely determined by f₀,f₁, and formulas for them are given.     

On certain decompositions of complex inverse Toeplitz matrices and related fast algorithms for solving linear systems with Toeplitz coefficient matrices

Formulas for inverting nonsingular Toeplitz matrices with complex entries are derived. These formulas either refine known ones or are new. They make it possible to develop economical algorithms for calculating products of inverse Toeplitz matrices with vectors.     

Toeplitz approximate inverse preconditioner for banded Toeplitz matrices

In this paper we introduce a new preconditioner for banded Toeplitz matrices, whose inverse is itself a Toeplitz matrix. Given a banded Hermitian positive definite Toeplitz matrixT, we construct a Toepliz matrixM such that the spectrum ofMT is clustered around one; specifically, if the bandwidth ofT is , all but eigenvalues ofMT are exactly one. Thus the preconditioned conjugate gradient method converges in +1 steps which is about half the iterations as required by other preconditioners for Toepliz systems that have been suggested in the literature. This idea has a natural extension to non-banded and non-Hermitian Toeplitz matrices, and to block Toeplitz matrices with Toeplitz blocks which arise in many two dimensional applications in signal processing. Convergence results are given for each scheme, as well as numerical experiments illustrating the good convergence properties of the new preconditioner.Partly supported by a travel fund from the Deutsche Forschungsgemeinschaft.Research supported in part by Oak Ridge Associated Universities grant no. 009707.