The conjugate-normal Toeplitz problem

The conjugate-normal Toeplitz problem is the one of characterizing the matrices that are conjugate-normal and Toeplitz at the same time. Based on a result of Gu and Patton and our results related to the normal Hankel problem, we show that a complex matrix is conjugate-normal and Toeplitz if and only if it is in one of the seven classes explicitly described in our paper.     

On tridiagonal conjugate-normal matrices

Conjugate-normal matrices play the same role in the theory of unitary congruences as conventional normal matrices do with respect to unitary similarities. Naturally, the properties of both matrix classes are fairly similar up to the distinction between the congruence and similarity. However, in certain respects, conjugate-normal matrices differ substantially from normal ones. Our goal in this paper is to indicate one of such distinctions. It is shown that none of the familiar characterizations of normal matrices having the irreducible tridiagonal form has a natural counterpart in the case of conjugate-normal matrices.     

Low-rank perturbations of symmetric matrices and their condensed forms under unitary congruences

The method MINRES-CN was earlier proposed by the authors for solving systems of linear equations with conjugate-normal coefficient matrices. It is now shown that this method is also applicable even if the coefficient matrix, albeit not conjugate-normal, is a low-rank perturbation of a symmetric matrix. If the perturbed matrix is still conjugate-normal, then, starting from some iteration step, the recursion underlying MINRES-CN becomes a three-term relation. These results are proved in terms of matrix condensed forms with respect to unitary congruences.     

Conjugate-normal matrices: A survey

Conjugate-normal matrices play the same important role in the theory of unitary congruence as the conventional normal matrices do with respect to unitary similarities. However, unlike the latter, the properties of conjugate-normal matrices are not widely known. Motivated by this fact, we give a survey of the properties of these matrices. In particular, a list of more than forty conditions is given, each of which is equivalent to A being conjugate-normal.     

On inversion of square matrices partitioned into non-square blocks

Inversion theorems for structured block matrices with non-square blocks are presented. The considered classes contain Toeplitz, Toeplitz plus Hankel and Van der Monde type matrices.     

New regions including eigenvalues of Toeplitz matrices

Two new eigenvalue inclusion regions for matrices with a constant main diagonal are given. We then apply these results to Toeplitz matrices, and obtain two regions including all eigenvalues of Toeplitz matrices. Furthermore, it is proved that the new regions are tighter than those in [Melman A. Ovals of Cassini for Toeplitz matrices, Linear and Multilinear Algebra. 2012;60:189–199].     

<Literal>smt</Literal>: a Matlab toolbox for structured matrices

The full exploitation of the structure of large scale algebraic problems is often crucial for their numerical solution. Matlab is a computational environment which supports sparse matrices, besides full ones, and allows one to add new types of variables (classes) and define the action of arithmetic operators and functions on them. The smt toolbox for Matlab introduces two new classes for circulant and Toeplitz matrices, and implements optimized storage and fast computational routines for them, transparently to the user. The toolbox, available in Netlib, is intended to be easily extensible, and provides a collection of test matrices and a function to compute three circulant preconditioners, to speed up iterative methods for linear systems. Moreover, it incorporates a simple device to add to the toolbox new routines for solving Toeplitz linear systems.     

Low-rank perturbations of normal and conjugate-normal matrices and their condensed forms under unitary similarities and congruences

Two theorems are proved on the condensed forms with respect to unitary similarity and congruence transformations. They provide a theoretical basis for constructing economical iterative methods for systems of linear equations whose matrices are low-rank perturbations of normal and conjugate-normal matrices.     

On the properties of matrices defining some classes of BVMs

Summary It is known that the matrices defining the discrete problem generated by a k-step Boundary Value Method (BVM) have a quasi-Toeplitz band structure [7]. In particular, when the boundary conditions are skipped, they become Toeplitz matrices. In this paper, by introducing a characterization of positive definiteness for such matrices, we shall prove that the Toeplitz matrices which arise when using the methods in the classes of BVMs known as Generalized BDF and Top Order Methods have such property. Mathematics Subject Classification (2000):65L06, 47B35, 15A48Work supported by G.N.C.S.     

On the skew-symmetric part of the Toeplitz component in the real normal (<Emphasis Type="Italic">T</Emphasis> + <Emphasis Type="Italic">H</Emphasis>)-problem

The real normal Toeplitz-plus-Hankel problem is to characterize the matrices that can be represented as sums of two real matrices of which one is Toeplitz and the other Hankel. For a matrix of this type, relations are found between the skew-symmetric part of the Toeplitz component and the matrix obtained by reversing the order of columns in the Hankel component.     

Monotone positive stable matrices

The class of real matrices which are both monotone (inverse positive) and positive stable is investigated. Such matrices, called N-matrices, have the well-known class of nonsingular M-matrices as a proper subset. Relationships between the classes of N-matrices, M-matrices, nonsingular totally nonnegative matrices, and oscillatory matrices are developed. Conditions are given for some classes of matrices, including tridiagonal and some Toeplitz matrices, to be N-matrices.     

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.     

On the coneigenvalues and singular values of a complex square matrix

It is shown that the coneigenvalues of a matrix, when properly defined (in a way different from the one commonly used in the literature), obey relations similar to the classical inequalities between the (ordinary) eigenvalues and singular values. Several interesting spectral properties of conjugate-normal matrices are indicated. This matrix class plays the same role in the theory of unitary congruences as the class of normal matrices plays in the theory of unitary similarities. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 334, 2006, pp. 111–120.     

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.     

A method of explicit factorization of matrix functions and applications

A method of explicit factorization of matrix functions of second order is proposed. The method consists of reduction of this problem to two scalar barrier problems and a finite system of linear equations. Applications to various classes of singular integral equations and equations with Toeplitz and Hankel matrices are given.     

Multivariate Polynomials,Duality, and Structured Matrices

We first review the basic properties of the well known classes of Toeplitz, Hankel, Vandermonde, and other related structured matrices and reexamine their correlation to operations with univariate polynomials. Then we define some natural extensions of such classes of matrices based on their correlation to multivariate polynomials. We describe the correlation in terms of the associated operators of multiplication in the polynomial ring and its dual space, which allows us to generalize these structures to the multivariate case. Multivariate Toeplitz, Hankel, and Vandermonde matrices, Bezoutians, algebraic residues, and relations between them are studied. Finally, we show some applications of this study to rootfinding problems for a system of multivariate polynomial equations, where the dual space, algebraic residues, Bezoutians, and other structured matrices play an important role. The developed techniques enable us to obtain a better insight into the major problems of multivariate polynomial computations and to improve substantially the known techniques of the study of these problems. In particular, we simplify and/or generalize the known reduction of the multivariate polynomial systems to the matrix eigenproblem, the derivation of the Bézout and Bernshtein bounds on the number of the roots, and the construction of multiplication tables. From the algorithmic and computational complexity point, we yield acceleration by one order of magnitude of the known methods for some fundamental problems of solving multivariate polynomial systems of equations.     

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.