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.     

Band matrices with Toeplitz inverses

It is shown that a square band matrix H=(h_ij) with h_ij=0 for j? i>r and i?j>s, where r+s is less than the order of the matrix, has a Toeplitz inverse if and only if it has a special structure characterized by two polynomials of degrees r and s, respectively.     

Toeplitz matrices with banded inverses

Given a Toeplitz matrix T with banded inverse [i.e., (T^?1)_ij=0 for j?i>p], we show that the elements of T can be expressed in terms of the roots of a polynomial. Then, using properties we have previously established, we generalize this result appropriately to allow singular T and show that the converse also holds. Finally, we give a sufficient condition for the decay of the elements of T as one moves away from the diagonal.     

Toeplitz matrices with totally nonnegative inverses

It is shown that the inverse of a Toeplitz matrix has only nonnegative minors if the zeros of a certain polynomial are positive or if their arguments are less than π?(k+n), where n is the dimension and k+1 is the bandwidth of the matrix.     

The inverses of Toeplitz band matrices

The elements of the inverse of a Toeplitz band matrix are given in terms ofthe solution of a difference equation. The expression for these elements is a quotient of determinants whose orders depend the number of nonzero superdiagonals but not on the order of the matrix. Thus, the formulae are particularly simple for lower triangular and lower Hessenberg Toeplitz matrices. When the number of nonzero superdiagonals is small, sufficient conditions on the solution of the abovementioned difference equation can be given to ensure that the inverse matrix is positive. If the inverse is positive, the row sums can be expressed in terms of the solution of the difference equation.     

Generalized inverses of certain Toeplitz matrices

The spectral inverse A^s of a Toeplitz matrix A whose form is related to that of a circulant matrix is studied by describing the algebraic structure of the semigroup of all matrices commuting with a given matrix with distinct eigenvalues. A computational form for A^s is given and necessary and sufficient conditions are found for A^s to be the Moore-Penrose generalized inverse A⁺.     

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.     

Eigenvalues of Toeplitz matrices associated with orthogonal polynomials

A connection between the asymptotic distribution of the zeros of orthogonal polynomials and the asymptotic behavior of the eigenvalues of Toeplitz matrices associated with these orthogonal polynomials is given. The result is applied to various families of orthogonal polynomials.     

On inverses and eigenpairs of periodic tridiagonal Toeplitz matrices with perturbed corners

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.     

Matrix orthogonal polynomials associated with perturbations of block Toeplitz matrices

Let {c _j } _j=0 ⁿ be a sequence of matrix moments associated with a matrix of measures supported on the unit circle, and let {P _j } _j=0 ⁿ be its corresponding sequence of monic matrix orthogonal polynomials. In this contribution, we consider a perturbation on the moments and find an explicit relation for the perturbed orthogonal polynomials in terms of {P _j } _j=0 ⁿ . We also obtain an expression for the corresponding second kind polynomials.     

Matrices with block Toeplitz inverses

Conditions for a nonsingular matrix to have a block Toeplitz inverse are obtained. A simpler criterion for a block Toeplitz matrix to have a block Toeplitz inverse is also given. The results generalize those of Huang and Cline for Toeplitz matrices with scalar entries, for which alternative statements and proofs are also indicated.     

Algorithms for finding the minimal polynomials and inverses of resultant matrices

In this paper, algorithms for computing the minimal polynomial and the common minimal polynomial of resultant matrices over any field are presented by means of the approach for the Gröbner basis of the ideal in the polynomial ring, respectively, and two algorithms for finding the inverses of such matrices are also presented. Finally, an algorithm for the inverse of partitioned matrix with resultant blocks over any field is given, which can be realized by CoCoA 4.0, an algebraic system over the field of rational numbers or the field of residue classes of modulo prime number. We get examples showing the effectiveness of the algorithms.     

Inversion of conjugate-Toeplitz matrices and conjugate-Hankel matrices

The inverses of conjugate-Toeplitz (CT) and conjugate-Hankel (CH) matrices can be expressed by the Gohberg–Heinig type formula. We obtain an explicit inverse formula of CT matrix. Similarly, the formula and the decomposition of the inverse of a CH matrix are provided. Also the stability of the inverse formulas of CT and CH matrices are discussed. Examples are provided to verify the feasibility of the algorithms.     

Toeplitz Preconditioners for Toeplitz Systems with Nonnegative Generating Functions

The preconditioned conjugate gradient method is employed tosolve Toeplitz systems T_nx = b where the generating functionsof the n-by-n Toeplitz matrices T_n are continuous nonnegativeperiodic functions defined in [–,]. The preconditionedC_n are band Toeplitz matrices with band-widths independent ofn. We prove that the spectra of C_n^-1T_n are uniformly boundedby constants independent of n. In particular, we show that thesolutions of T_nx = b can be obtained in O(nlogn) operations.     

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

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

.     

Determinantal formulae for matrices with sparse inverses

The determinant of a matrix is expressed in terms of certain of its principal minors by a formula which can be “read off” from the graph of the inverse of the matrix. The only information used is the zero pattern of the inverse, and each zero pattern yields one or more corresponding formulae for the determinant.     

On five-diagonal Toeplitz matrices and orthogonal polynomials on the unit circle

This paper deals with modifications of the Lebesgue moment functional by trigonometric polynomials of degree 2 and their associated orthogonal polynomials on the unit circle. We use techniques of five-diagonal matrix factorization and matrix polynomials to study the existence of such orthogonal polynomials.Dedicated to Prof. Luigi Gatteschi on his 70th birthdayThis research was partially supported by Diputación General de Aragón under grant P CB-12/91.     

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.     

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.     

Nonsymmetric Toeplitz matrices that commute with tridiagonal matrices

Translated from Matematicheskie Zametki, Vol. 55, No. 5, pp. 69–79, May, 1994.