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.     

关于Toeplitz矩阵的某些注记

In this paper,we study real symmetric Toeplitz matrices commutable with tridi-agonal matrices, present more detailed results than those in [1], and extend them to non-symmetric Toeplitz matrices. Also, complex Toeplitz matrices, especially the corresponding matrices of lower order, are discussed.     

Block toeplitz products of block toeplitz matrices

Necessary and sufficient conditions for the product of two block Toeplitz matrices to be block Toeplitz are obtained. In the special case of two Toeplitz matrices, the conditions simplify considerably and, when combined with known necessary and sufficient conditions for a nonsingular Toeplitz matrix to have a Toeplitz inverse, provide a simple characterization of the additional matrix structure required by a subclass of Toeplitz matrices in order for it to be closed with respect to both inversion and multiplication.     

BANDED TOEPLITZ PRECONDITIONERS FOR TOEPLITZ MATRICES FROM SINC METHODS

We give general expressions, analyze algebraic properties and derive eigenvalue bounds for a sequence of Toeplitz matrices associated with the sinc discretizations of various orders of differential operators. We demonstrate that these Toeplitz matrices can be satisfactorily preconditioned by certain banded Toeplitz matrices through showing that the spectra of the preconditioned matrices are uniformly bounded. In particular, we also derive eigenvalue bounds for the banded Toeplitz preconditioners. These results are elementary in constructing high-quality structured preconditioners for the systems of linear equations arising from the sinc discretizations of ordinary and partial differential equations, and are useful in analyzing algebraic properties and deriving eigenvalue bounds for the corresponding preconditioned matrices. Numerical examples are given to show effectiveness of the banded Toeplitz preconditioners.     

一类Toeplitz矩阵的群逆

Ｔｏｅｐｌｉｔｚ矩阵Ｔｎ＝（ｔｉ－ｊ）ｎｉ，ｊ＝０在信号处理、系统理论、逼近论、正交多项式、积分方程数值解等许多领域常常遇到．易知，Ｔｏｅｐｌｉｔｚ矩阵Ｔｎ的逆矩阵一般不再是Ｔｏｅｐｌｉｔｚ矩阵．１９７２年，Ｇｏｈｂｅｒｇ和Ｓｅｍｅｎｃｕｌ［１］给出...     

Extremal Trigonometric and Power Polynomials in Several Variables

We study extremal nonnegative polynomials in several variables. Our approach makes substantial use of block Toeplitz matrices. Note that the blocks of these matrices are themselves Toeplitz 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].     

Spectral asymptotics for Kac–Murdock–Szegő matrices

Szeg?’s First Limit Theorem provides the limiting statistical distribution of the eigenvalues of large Toeplitz matrices. Szeg?’s Second (or Strong) Limit Theorem for Toeplitz matrices gives a second order correction to the First Limit Theorem, and allows one to calculate asymptotics for the determinants of large Toeplitz matrices. In this paper we survey results extending the First and Second Limit Theorems to Kac–Murdock–Szeg? (KMS) matrices. These are matrices whose entries along the diagonals are not necessarily constants, but modeled by functions. We clarify and extend some existing results, and explain some apparently contradictory results in the literature.     

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.     

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.     

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.     

块Toeplitz方程组的快速块Gauss-Seidel迭代算法

本文研究块Toeplitz方程组的块Gauss-Seidel迭代算法。我们首先讨论了块三角Toeplitz矩阵的一些性质,然后给出了求解块三角Toeplitz矩阵逆的快速算法,由此而得到了求解块Toeplitz方程组的快速块Gauss-Seidel迭代算法,最后证明了当系数矩阵为对称正定和H-矩阵时该方法都收敛,数值例子验证了方法的收敛性。     

Band Toeplitz preconditioners for block Toeplitz systems

We consider the solutions of block Toeplitz systems with Toeplitz blocks by the preconditioned conjugate gradient (PCG) method. Here the block Toeplitz matrices are generated by nonnegative functions f(x,y). We use band Toeplitz matrices as preconditioners. The generating functions g(x,y) of the preconditioners are trigonometric polynomials of fixed degree and are determined by minimizing (f − g)/f∞. We prove that the condition number of the preconditioned system is O(1). An a priori bound on the number of iterations for convergence is obtained.     

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.     

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.     

Some extensions of Hankel and Toeplitz matrices

New necessary and sufficient conditions are given for a nonsingular matrix to have Hankel or Toeplitz form. Two generalizations, termed conjugate-Toeplitz and Leslie-Toeplitz matrices, are introduced A number of properties of Toeplitz matrices are shown to carry over to these extensions     

Riordan arrays and the LDU decomposition of symmetric Toeplitz plus Hankel matrices

We examine a result of Basor and Ehrhardt concerning Hankel and Toeplitz plus Hankel matrices, within the context of the Riordan group of lower-triangular matrices. This allows us to determine the LDU decomposition of certain symmetric Toeplitz plus Hankel matrices. We also determine the generating functions and Hankel transforms of associated sequences.     

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.