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.     

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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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.     

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.     

Infinite extensions of Toeplitz matrices

A formula for the smallest of the ranks of infinite Toeplitz extensions of a finite rectangular Toeplitz matrix is derived. Bibliography: 3 titles._________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 296, 2003, pp. 5–14.     

QR factorization of Toeplitz matrices

Summary This paper presents a new algorithm for computing theQR factorization of anm×n Toeplitz matrix inO(mn) operations. The algorithm exploits the procedure for the rank-1 modification and the fact that both principal (m–1)×(n–1) submatrices of the Toeplitz matrix are identical. An efficient parallel implementation of the algorithm is possible.     

Nonsymmetric Toeplitz matrices that commute with tridiagonal matrices

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

Functions generating normal Toeplitz matrices

To any complex function there corresponds a Fourier series, which is often associated with a sequence {T _n} of Toeplitz n × n matrices. Functions whose Fourier series generate sequences of normal Toeplitz matrices are classified, and a procedure for constructing Fourier series for which the sequence {T _n} contains an infinite subsequence of normal matrices is described.     

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.     

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⁺.     

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.     

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.     

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.     

The conditioning of Toeplitz band matrices

In this paper, practical conditions to check the well-conditioning of a family of nonsingular Toeplitz band matrices are obtained. All the results are based on the location of the zeros of a polynomial associated with the given family of Toeplitz matrices.The same analysis is also used to derive uniform componentwise bounds for the entries of the inverse matrices in such family.     

On algebras of Toeplitz fuzzy matrices

In this paper, necessary and sufficient conditions are given for a product of Toeplitz fuzzy matrices to be Toeplitz. As an application, a criterion for normality of Toeplitz fuzzy matrices is derived and conditions are deduced for symmetric idempotency of Toeplitz fuzzy matrices. We discuss similar results for Hankel fuzzy matrices. Keywords: Fuzzy matrix, Toeplitz and Hankel matrices.