n元多项式矩阵有MLP分解的新判别算法

基于Lin-Bose猜想,研究l×m阶矩阵MLP分解问题,利用l×m阶矩阵的2×2级子式、矩阵的元素来刻画矩阵MLP分解的条件,得到一些有趣的结果,并给出多元多项式矩阵有MLP分解的新判别算法.     

整系数多项式的"分解矩阵"及其应用

在数学上 ,求微分方程的特征根、矩阵的特征值时 ,都会遇到多项式的因式分解问题 ;在工程上 ,研究动态系统的稳定性等问题时 ,也会遇到多项式的因式分解问题。传统的因式分解法有一定的局限性 ,它只适合于一些低次多项式或较规则的高次多项式的分解 ,而对一般高次多项式的因式分解 ,传统的方法常显出它的缺陷。本文就整系数多项式的因式分解问题 ,给出了一个比较好用的方法——矩阵法。该方法的核心就是根据多项式构造一个“分解矩阵”,再用此“分解矩阵”对多项式进行因式分解。该方法具有简便、实用的特点 ,特别适用于高次多项式的因式分…     

构造一般Dixon 结式矩阵的快速算法

近几年来, 基于Dixon 结式的消去法被广泛地用来求解非线性多项式方程组, 因此国际上许多学者开始研究构造Dixon 结式矩阵的有效算法. 本文将目前最为有效的只能处理2个变元3个方程情形的递归算法扩展到n个变元n+1个多项式方程的一般情形, 并且将该算法在Maple下编程实现. 通过Maple随机产生的多项式的比较实验, 可以看出, 比之现有的所有方法, 本程序具有更高的效率. 特别是应用此程序, 首次以48阶的Dixon结式矩阵的形式, 给出了4个一般的关于每个变元的次数不超过2次的曲面存在公共交点的必要条件.     

矩阵函数f(A)的计算方法

通过以λ为变量的多项式f(λ)定义了矩阵多项式f(A),并将矩阵多项式的计算方法推广到矩阵函数.同时给出了矩阵函数f(A)的又一种计算方法.     

一般基下矩阵多项式的张量Bezoutian

给出矩阵多项式在一般基下的张量Bezoutian的定义,推广了标准幂基下的古典张量Bezoutian.讨论了该矩阵的Barnett型分解,缠绕关系和关于可控制/可观测矩阵的表示等重要性质.     

两类下三角形Pascal矩阵的相似性

本文研究了Pascal矩阵与位移Pascal矩阵之间的关系.利用组合恒等式与矩阵分解的方法,得到了Pascal矩阵以及位移Pascal矩阵与若当标准型之间的过渡矩阵.同时也得到了这两类矩阵在域Zp上的最小多项式.     

有限域上正交循环矩阵的构造和计数

<正>交循环矩阵在组合、编码等诸多研究领域具有重要的应用价值.运用提升理想和傅里叶反演变换等方法,给出了任意有限域上任意阶正交循环矩阵的构造方法,以及精确的计数结果.区别于已有的构造方法,此提升方法不需要分解模多项式.     

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.     

一种改进的线性系统的有限时间平衡截断（英文）

本文提出一种改进的线性系统的有限时间平衡截断方法.该方法首先利用Shifted Legendre多项式对线性系统的有限时间可控Gram矩阵和可观Gram矩阵进行近似低秩分解,其中根据正交多项式与幂级数之间的关系,该近似低秩分解因子可以通过简单的递推公式得到,然后构造正交投影变换得到近似平衡系统,进而通过截断较小的Hankel奇异值对应的状态得到降阶系统.此外,本文还简要讨论了该降阶模型的稳定性.最后,通过数值算例验证了算法的有效性.     

一个特殊的二项系数

利用发生函数和矩阵方法,研究了一个特殊的二项式系数[n λ n-k]和它所构成的矩阵.得到以[n λ n-k]为矩阵元素的Pascal型矩阵的指数分解和乘积分解公式.同时,考察了与二项式型多项式相伴的函数矩阵Pn,λ[x]及其性质.     

Generalized Leibniz functional matrices and factorizations of some well-known matrices

In this paper, we introduce the generalized Leibniz functional matrices and study some algebraic properties of such matrices. To demonstrate applications of these properties, we derive several novel factorization forms of some well-known matrices, such as the complete symmetric polynomial matrix and the elementary symmetric polynomial matrix. In addition, by applying factorizations of the generalized Leibniz functional matrices, we redevelop the known results of factorizations of Stirling matrices of the first and second kind and the generalized Pascal matrix.     

Stable factorizations of monic matrix polynomials and stable invariant subspaces

The stable factorizations of a monic matrix polynomial are characterized in terms of spectral properties. Proofs are based on the divisibility theory developed by I. Gohberg, P. Lancaster and L. Rodman. A large part of the paper is devoted to a detailed analysis of stable invariant subspaces of a matrix. The results are also used to describe all stable solutions of the operator Riccati equation.     

To solving problems of algebra for two-parameter matrices. VII

The paper discusses the method of rank factorization for solving spectral problems for two-parameter polynomial matrices. New forms of rank factorization, which are computed using unimodular matrices only, are suggested. Applications of these factorizations to solving spectral problems for two-parameter polynomial matrices of both general and special forms are presented. In particular, matrices free of the singular spectrum are considered. Conditions sufficient for a matrix to be free of the singular spectrum and also conditions sufficient for a basis matrix of the null-space to have neither the finite regular nor the finite singular spectrum are provided. Bibliography: 3 titles.     

Factorization of matrix polynomials with multiple roots

The concept of a multiple root of matrix polynomial L(λ) is introduced, and associated spectral properties of L(λ) are investigated. A statement concerning factorization of L(λ) is presented. Applications are made to factorizations of the matrix polynomial L^α(λ), for any positive integer α.     

Factorization of multivariate positive Laurent polynomials

Recently Dritschel proved that any positive multivariate Laurent polynomial can be factorized into a sum of square magnitudes of polynomials. We first give another proof of the Dritschel theorem. Our proof is based on the univariate matrix Fejér–Riesz theorem. Then we discuss a computational method to find approximates of polynomial matrix factorization. Some numerical examples will be shown. Finally we discuss how to compute nonnegative Laurent polynomial factorizations in the multivariate setting.     

Parallel factorizations of polynomial matrices

Conditions are established under which suggested factorizations of polynomial matrices over a field are parallel to factorizations of their canonical diagonal forms. An existence criterion of these factorizations of polynomial matrices is indicated and a method of constructing them is suggested.Translated from Ukrayins'kyy Matematychnyy Zhurnal, Vol. 44, No. 9, pp. 1228–1233, September, 1992.     

紧支撑正交的二维小波

基于Householder矩阵扩充,构造了紧支撑正交的二维小波,所构造小波函数的支撑不超过尺度函数的支撑,并且给出了容易实施的显式构造算法．另外,还通过构造反例说明Riesz定理不适用于二元三角多项式．最后,构造了算例．     

Bivariate Polynomial Natural Spline Interpolation Algorithms with Local Basis for Scattered Data

Because of its importance in both theory and applications, multivariate splines have attracted special attention in many fields. Based on the theory of spline functions in Hilbert spaces, bivariate polynomial natural splines for interpolating, smoothing or generalized interpolating of scattered data over an arbitrary domain are constructed with one-sided functions. However, this method is not well suited for large scale numerical applications. In this paper, a new locally supported basis for the bivariate polynomial natural spline space is constructed. Some properties of this basis are also discussed. Methods to order scattered data are shown and algorithms for bivariate polynomial natural spline interpolating are constructed. The interpolating coefficient matrix is sparse, and thus, the algorithms can be easily implemented in a computer.     

To Solving Multiparameter Problems of Algebra. 5. The ∇V-q Factorization Algorithm and its Applications

The algorithm of ∇V-factorization, suggested earlier for decomposing one- and two-parameter polynomial matrices of full row rank into a product of two matrices (a regular one, whose spectrum coincides with the finite regular spectrum of the original matrix, and a matrix of full row rank, whose singular spectrum coincides with the singular spectrum of the original matrix, whereas the regular spectrum is empty), is extended to the case of q-parameter (q ≥ 1) polynomial matrices. The algorithm of ∇V-q factorization is described, and its justification and properties for matrices with arbitrary number of parameters are presented. Applications of the algorithm to computing irreducible factorizations of q-parameter matrices, to determining a free basis of the null-space of polynomial solutions of a matrix, and to finding matrix divisors corresponding to divisors of its characteristic polynomial are considered. Bibliogrhaphy: 4 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 309, 2004, pp. 144–153.