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.     

Structure theory and fast inversion of Hankel striped matrices. I

A structure theory for Hankel striped matrices is developed that generalizes the corresponding theory for Hankel matrices [4]. This leads, in particular, to an inverslon algorithm for Hankel and Toeplitz striped matrices working without additional assumptions.     

On two particular cases of solving the normal Hankel problem

The normal Hankel problem is one of characterizing all the complex matrices that are normal and Hankel at the same time. The matrix classes that can contain normal Hankel matrices admit a parameterization by real 2 × 2 matrices with determinant one. Here, the normal Hankel problem is solved in the case where the characteristic matrix of a given class is an order two Jordan block for the eigenvalue 1 or ?1.     

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.     

Generalized Hankel matrices of Markov parameters and their applications to control problems

The concept of Hankel matrices of Markov parameters associated with two polynomials is generalized for matrices. The generalized Hankel matrices of Markov parameters are then used to develop methods for testing the relative primeness of two matrices A and B, for determining stability and inertia of a matrix, and for constructing a class of matrices C such that A + C has a desired spectrum. Neither the method of construction of the generalized Hankel matrices nor the methods developed using these matrices require explicit computation of the characteristic polynomial of A (or of B).     

由特征值唯一确定的实3×3Hankel矩阵(英文)

本文研究了由特征值唯一确定的3×3实Hankel矩阵.借助于M.Fielder[1]的结论并经过细致的讨论,得到3×3实Hankel矩阵由其特征值唯一确定的充分必要条件,刻画了3×3实Hankel矩阵的一种特征值性质.     

支撑在(-1, 1)上的Carleson型测度与Hankel 矩阵

In this paper, we establish a connection between Carleson type measures supported on(-1, 1) and certain Hankel matrices. The connection is given by the study of Hankel matrices acting on Dirichlet type spaces.     

Quasidirect decompositions of Hankel and Toeplitz matrices

Decompositions, over an algebraically closed field, of a Hankel matrix into a sum of Hankel matrices the sum of the ranks of which is equal to the rank of the original matrix, are completely described. Similar results hold for Toeplitz matrices.     

Block diagonalization and LU-equivalence of Hankel matrices

This article presents a new algorithm for obtaining a block diagonalization of Hankel matrices by means of truncated polynomial divisions, such that every block is a lower Hankel matrix. In fact, the algorithm generates a block LU-factorization of the matrix. Two applications of this algorithm are also presented. By the one hand, this algorithm yields an algebraic proof of Frobenius’ Theorem, which gives the signature of a real regular Hankel matrix by using the signs of its principal leading minors. On the other hand, the close relationship between Hankel matrices and linearly recurrent sequences leads to a comparison with the Berlekamp–Massey algorithm.     

On the reduction of the normal Hankel problem to two particular cases

The normal Hankel problem (NHP) is to describe complex matrices that are normal and Hankel at the same time. The available results related to the NHP can be combined into two groups. On the one hand, there are several known classes of normal Hankel matrices. On the other hand, the matrix classes that may contain normal Hankel matrices not belonging to the known classes were shown to admit a parametrization by real 2 × 2 matrices with determinant 1. We solve the NHP for the cases where the characteristic matrix W of the given class has: (a) complex conjugate eigenvalues; (b) distinct real eigenvalues. To obtain a complete solution of the NHP, it remains to analyze two situations: (1) W is the Jordan block of order two for the eigenvalue 1; (2) W is the Jordan block of order two for ?1.     

满秩Hankel矩阵的拟直分解

本文研究行满秩Hankel矩阵分解为一个真正的(proper)Hankel矩阵与一个退化的(de- generate)Hankel矩阵之拟直和的存在性及唯一性问题.     

Asymptotic Formulas for Determinants of a Sum of Finite Toeplitz and Hankel Matrices

The purpose of this paper is to describe asymptotic formulas for determinants of a sum of finite Toeplitz and Hankel matrices with singular generating functions. The formulas are similar to those of the analogous problem for finite Toeplitz matrices for a certain class of symbols. However, the appearance of the Hankel matrices changes the nature of the asymptotics in some instances depending on the location of the singularities. Several concrete examples are also described in the paper.     

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.     

A complete solution of the normal Hankel problem

The normal Hankel problem is the one of characterizing the matrices that are normal and Hankel at the same time. We give a complete solution of this problem.     

Analysis of the Structured Total Least Squares Problem for Hankel/Toeplitz Matrices

The Structured Total Least Squares (STLS) problem is a natural extension of the Total Least Squares (TLS) approach when structured matrices are involved and a similarly structured rank deficient approximation of that matrix is desired. In many of those cases the STLS approach yields a Maximum Likelihood (ML) estimate as opposed to, e.g., TLS.In this paper we analyze the STLS problem for Hankel matrices (the theory can be extended in a straightforward way to Toeplitz matrices, block Hankel and block Toeplitz matrices). Using a particular parametrisation of rank-deficient Hankel matrices, we show that this STLS problem suffers from multiple local minima, the properties of which depend on the parameters of the new parametrisation. The latter observation makes initial estimates an important issue in STLS problems and a new initialization method is proposed. The new initialization method is applied to a speech compression example and the results confirm the improved performance compared to other previously proposed initialization methods.     

Hankel matrices for the period-doubling sequence

We give an explicit evaluation, in terms of products of Jacobsthal numbers, of the Hankel determinants of order a power of two for the period-doubling sequence. We also explicitly give the eigenvalues and eigenvectors of the corresponding Hankel matrices. Similar considerations give the Hankel determinants for other orders.     

Minimal condition number for positive definite Hankel matrices using semidefinite programming

We present a semidefinite programming approach for computing optimally conditioned positive definite Hankel matrices of order n. Unlike previous approaches, our method is guaranteed to find an optimally conditioned positive definite Hankel matrix within any desired tolerance. Since the condition number of such matrices grows exponentially with n, this is a very good test problem for checking the numerical accuracy of semidefinite programming solvers. Our tests show that semidefinite programming solvers using fixed double precision arithmetic are not able to solve problems with n>30. Moreover, the accuracy of the results for 24?n?30 is questionable. In order to accurately compute minimal condition number positive definite Hankel matrices of higher order, we use a Mathematica 6.0 implementation of the SDPHA solver that performs the numerical calculations in arbitrary precision arithmetic. By using this code, we have validated the results obtained by standard codes for n?24, and we have found optimally conditioned positive definite Hankel matrices up to n=100.