An extended version of Schur-Cohn-Fujiwara theorem in stability theory

This paper is concerned with root localization of a complex polynomial with respect to the unit circle in the more general case. The classical Schur-Cohn-Fujiwara theorem converts the inertia problem of a polynomial to that of an appropriate Hermitian matrix under the condition that the associated Bezout matrix is nonsingular. To complete it, we discuss an extended version of the Schur-Cohn-Fujiwara theorem to the singular case of that Bezout matrix. Our method is mainly based on a perturbation technique for a Bezout matrix. As an application of these results and methods, we further obtain an explicit formula for the number of roots of a polynomial located on the upper half part of the unit circle as well.     

关于矩阵多项式的逆矩阵求法的一个注记

给出了矩阵多项式逆矩阵的一些充要条件和一种求法.     

A finite algorithm for the Drazin inverse of a polynomial matrix

Based on Greville's finite algorithm for Drazin inverse of a constant matrix we propose a finite numerical algorithm for the Drazin inverse of polynomial matrices. We also present a new proof for Decell's finite algorithm through Greville's finite algorithm.     

两类循环分块矩阵及其有关算法

本文利用多项式矩阵最大右公因式，给出R-循环分块矩阵的和对称R-循环分块矩阵非奇异以及线性方程组反问题有唯一解的充要条件，进而得到它们求逆、线性方程组唯一解、线性方程组在循环分块矩阵中的反总问题求唯一解的算法。     

多项式矩阵求逆的连分式插值方法

本文借助于基于广义逆矩阵Thiele-型连分式插值的计算公式，建立了多项式矩阵求逆的一个新方法。关于多项式矩阵求逆的一个实例给出以说明本文的结果。     

The inverse of circulant matrix

A direct method is proposed to get the inverse matrix of circulant matrix that find important application in engineering, the elements of the inverse matrix are functions of zero points of the characteristic polynomial g(z) and g′(z) of circulant matrix, four examples to get the inverse matrix are presented in the paper.     

一类非线性矩阵方程对称解的双迭代算法

利用逆矩阵的Neumann级数形式,将在Schur插值问题中遇到的含未知矩阵二次项之逆的非线性矩阵方程转化为高次多项式矩阵方程,然后采用牛顿算法求高次多项式矩阵方程的对称解,并采用修正共轭梯度法求由牛顿算法每一步迭代计算导出的线性矩阵方程的对称解或者对称最小二乘解,建立求非线性矩阵方程的对称解的双迭代算法.双迭代算法仅要求非线性矩阵方程有对称解,不要求它的对称解唯一,也不对它的系数矩阵做附加限定.数值算例表明,双迭代算法是有效的.     

矩阵多项式的逆矩阵的求法

给出了矩阵多项式的逆矩阵的一般求法.     

可逆矩阵的高次伴随矩阵

用数学归纳法推出了可逆矩阵的高次伴随矩阵的公式,并结合可逆矩阵的基本公式得出了可逆矩阵的高次伴随矩阵的行列式和逆矩阵,给出了可逆矩阵的高次伴随矩阵的特征值和特征向量的表示公式,最后讨论了若干个可逆矩阵的乘积的高次伴随矩阵.     

矩阵多项式秩的和的恒等式及其应用

证明了矩阵A的两个多项式秩的和等于它们最大公因式与最小公倍式秩的和,这个结果不仅可以概括近期文献的相关工作,而且可以对应用矩阵多项式求逆矩阵的方法作进一步的研究,同时也可使关于矩阵秩恒等式的最新讨论获得一种简单统一的处理方法.     

二次自伴矩阵多项式阵的特征值

系统地论证了二次自伴矩阵多项式特征值,特征向量的性质.给出了二次自伴矩阵多项式特征值与任一非零向量所对应的二次多项式根之间的大小关系;精确地给出了二次自伴矩阵多项式是负定时参数的界;简化了二次自伴矩阵多项式的符号特征是正(负)的特征值对应特征向量间可以是线性无关等定理的证明.     

求鳞状循环因子矩阵的极小多项式的算法

提出了任意域上鳞状循环因子矩阵 ,利用多项式环的理想的Go bner基的算法给出了任意域上鳞状循环因子矩阵的极小多项式和公共极小多项式的一种算法 .同时给出了这类矩阵逆矩阵的一种求法 .在有理数域或模素数剩余类域上 ,这一算法可由代数系统软件Co CoA4 .0实现 .数值例子说明了算法的有效性     

包含切比雪夫多项式的循环矩阵行列式的计算

行首加r尾r右循环矩阵和行尾加r首r左循环矩阵是两种特殊类型的矩阵,这篇论文中就是利用多项式因式分解的逆变换这一重要的技巧以及这类循环矩阵漂亮的结构和切比雪夫多项式的特殊的结构,分别讨论了第一类、第二类切比雪夫多项式的关于行首加r尾r右循环矩阵和行尾加r首r左循环矩阵的行列式,从而给出了行首加r尾r右循环矩阵和行尾加r首r左循环矩阵的行列式显式表达式.这些显式表达式与切比雪夫多项式以及参数r有关.这一问题的应用背景主要在循环编码,图像处理等信息理论方面.     

Radicals of matrix rings generated by a companion matrix

The radicals of full matrix rings as well as structural matrix rings have been studied extensively and there is a well-developed theory for both. Here we initiate the radical theory for another class of matrix rings over a given ring; the matrix ring generated by the companion matrix of a polynomial over the ring.     

Inverse problems for periodic generalized Jacobi matrices

Some inverse problems for semi-infinite periodic generalized Jacobi matrices are considered. In particular, a generalization of the Abel criterion is presented. The approach is based on the fact that the solvability of the Pell-Abel equation is equivalent to the existence of a certainly normalized J-unitary 2×2-matrix polynomial (the monodromy matrix).     

Newton-Like Iteration Based on a Cubic Polynomial for Structured Matrices

We recall Newtons iteration for computing the inverse or Moore–Penrose generalized inverse of a matrix. Then we specialize this approach to the case of structured matrices where all input, output and intermediate auxiliary matrices are represented in a compressed form, via their short displacement generators. We design a new Newton-like iteration based on a cubic polynomial and show its effectiveness by some numerical experiments for matrices from the Toeplitz-like class and the Cauchy-like class.     

Direct and inverse polynomial perturbations of hermitian linear functionals

This paper is devoted to the study of direct and inverse (Laurent) polynomial modifications of moment functionals on the unit circle, i.e., associated with hermitian Toeplitz matrices. We present a new approach which allows us to study polynomial modifications of arbitrary degree.The main objective is the characterization of the quasi-definiteness of the functionals involved in the problem in terms of a difference equation relating the corresponding Schur parameters. The results are presented in the general framework of (non-necessarily quasi-definite) hermitian functionals, so that the maximum number of orthogonal polynomials is characterized by the number of consistent steps of an algorithm based on the referred recurrence for the Schur parameters.The non-uniqueness of the inverse problem makes it more interesting than the direct one. Due to this reason, special attention is paid to the inverse modification, showing that different approaches are possible depending on the data about the polynomial modification at hand. These different approaches are translated as different kinds of initial conditions for the related inverse algorithm.Some concrete applications to the study of orthogonal polynomials on the unit circle show the effectiveness of this new approach: an exhaustive and instructive analysis of the functionals coming from a general inverse polynomial perturbation of degree one for the Lebesgue measure; the classification of those pairs of orthogonal polynomials connected by a certain type of linear relation with constant polynomial coefficients; and the determination of those orthogonal polynomials whose associated ones are related to a degree one polynomial modification of the original orthogonality functional.     

A recursive algorithm for constructing complicated Dixon matrices

Whether the determinant of the Dixon matrix equals zero or not is used for determining if a system of n + 1 polynomial equations in n variables has a common root, and is a very efficient quantifier elimination approach too. But for a complicated polynomial system, it is not easy to construct the Dixon matrix. In this paper, a recursive algorithm to construct the Dixon matrix is proposed by which some problems that cannot be tackled by other methods can be solved on the same computer platform. A dynamic programming algorithm based on the recursive formula is developed and compared for speed and efficiency to the recursive algorithm.