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.     

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.     

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.     

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.     

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.     

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.     

Algebras in matrix spaces with displacement structure

We introduce the problem of the location of the algebras contained in a matrix space with displacement structure. We present some partial solutions of the problem that unify and generalize various results obtained so far for the spaces of Toeplitz, Loewner and Toeplitz plus Hankel matrices.     

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.     

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.     

Tangential Nevanlinna-Pick Interpolation and Its Connection with Hamburger Matrix Moment Problem

This paper is concerned with the solution of a certain tangential Nevanlinna-Pick interpolation for Nevanlinna functions. We use the so-called block Hankel vector method to establish two intrinsic connections between the tangential Nevanlinna-Pick interpolation in the Nevanlinna class and the truncated Hamburger matrix moment problem associated with the block Hankel vector under consideration: one is a congruent relationship between their information matrices, and the other is a divisor-remainder connection between their solutions. These investigations generalize our previous work on the Nevanlinna-Pick interpolation and power matrix moment problem.     

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 fast Lanczos method for computation of eigenvalues of Hankel matrices using multiprecision arithmetics

The use of the fast Fourier transform (FFT) accelerates Lanczos tridiagonalisation method for Hankel and Toeplitz matrices by reducing the complexity of matrix–vector multiplication. In multiprecision arithmetics, the FFT has overheads that make it less competitive compared with alternative methods when the accuracy is over 10000 decimal places. We studied two alternative Hankel matrix–vector multiplication methods based on multiprecision number decomposition and recursive Karatsuba‐like multiplication, respectively. The first method was uncompetitive because of huge precision losses, while the second turned out to be five to 14 times faster than FFT in the ranges of matrix sizes up to n = 8192 and working precision of b = 32768 bits we were interested in. We successfully applied our approach to eigenvalues calculations to studies of spectra of matrices that arise in research on Riemann zeta function. The recursive matrix–vector multiplication significantly outperformed both the FFT and the traditional multiplication in these studies. Copyright © 2016 John Wiley & Sons, Ltd.     

Hankel matrices for system identification

The coefficients of a linear system, even if it is a part of a block-oriented nonlinear system, normally satisfy some linear algebraic equations via Hankel matrices composed of impulse responses or correlation functions. In order to determine or to estimate the coefficients of a linear system it is important to require the associated Hankel matrix be of row-full-rank. The paper first discusses the equivalent conditions for identifiability of the system. Then, it is shown that the row-full-rank of the Hankel matrix composed of impulse responses is equivalent to identifiability of the system. Finally, for the row-full-rank of the Hankel matrix composed of correlation functions, the necessary and sufficient conditions are presented, which appear slightly stronger than the identifiability condition. In comparison with existing results, here the minimum phase condition is no longer required for the case where the dimension of the system input and output is the same, though the paper does not make such a dimensional restriction.     

Fast algorithm for solving the Hankel/Toeplitz Structured Total Least Squares problem

The Structured Total Least Squares (STLS) problem is a natural extension of the Total Least Squares (TLS) problem when constraints on the matrix structure need to be imposed. Similar to the ordinary TLS approach, the STLS approach can be used to determine the parameter vector of a linear model, given some noisy measurements. In many signal processing applications, the imposition of this matrix structure constraint is necessary for obtaining Maximum Likelihood (ML) estimates of the parameter vector. In this paper we consider the Toeplitz (Hankel) STLS problem (i.e., an STLS problem in which the Toeplitz (Hankel) structure needs to be preserved). A fast implementation of an algorithm for solving this frequently occurring STLS problem is proposed. The increased efficiency is obtained by exploiting the low displacement rank of the involved matrices and the sparsity of the associated generators. The fast implementation is compared to two other implementations of algorithms for solving the Toeplitz (Hankel) STLS problem. The comparison is carried out on a recently proposed speech compression scheme. The numerical results confirm the high efficiency of the newly proposed fast implementation: the straightforward implementations have a complexity of O((m+n)³) and O(m³) whereas the proposed implementation has a complexity of O(mn+n²). This revised version was published online in June 2006 with corrections to the Cover Date.     

A fast method to block-diagonalize a Hankel matrix

In this paper, we consider an approximate block diagonalization algorithm of an n×n real Hankel matrix in which the successive transformation matrices are upper triangular Toeplitz matrices, and propose a new fast approach to compute the factorization in O(n ²) operations. This method consists on using the revised Bini method (Lin et al., Theor Comp Sci 315: 511–523, 2004). To motivate our approach, we also propose an approximate factorization variant of the customary fast method based on Schur complementation adapted to the n×n real Hankel matrix. All algorithms have been implemented in Matlab and numerical results are included to illustrate the effectiveness of our approach.     

标准混合差集矩阵的构造

通过利用差集矩阵和投影矩阵的正交分解之间的关系,首先提出了构造小的标准混合差集矩阵的一般方法.其次,给定一个阶为r+1的标准混合差集矩阵和一个阶为r的差集矩阵,首先提出了构造阶为r(r+1)的标准混合差集矩阵的一般方法.如果阶为r的差集矩阵不存在但一个试验次数为r~2的正交表存在,也可以通过它们构造阶数较大的标准混合差集矩阵.     

保持交换环上两类矩阵运算的映射

主要针对交换环上两类矩阵的保持问题进行展开:(1)刻画了交换环上全矩阵空间和上三角形矩阵空间的保持反对合矩阵映射的形式.(2)研究了交换环上n阶上三角形矩阵空间的保持伴随矩阵映射的形式.     

Every Matrix is a Product of Toeplitz Matrices

We show that every \(n\,\times \,n\) matrix is generically a product of \(\lfloor n/2 \rfloor + 1\) Toeplitz matrices and always a product of at most \(2n+5\) Toeplitz matrices. The same result holds true if the word ‘Toeplitz’ is replaced by ‘Hankel,’ and the generic bound \(\lfloor n/2 \rfloor + 1\) is sharp. We will see that these decompositions into Toeplitz or Hankel factors are unusual: We may not, in general, replace the subspace of Toeplitz or Hankel matrices by an arbitrary \((2n-1)\)-dimensional subspace of \({n\,\times \,n}\) matrices. Furthermore, such decompositions do not exist if we require the factors to be symmetric Toeplitz or persymmetric Hankel, even if we allow an infinite number of factors.