关于随机矩阵Kronecker积的谱半径的不等式

研究了随机矩阵的Kronecker积的数学期望的性质,得到了随机矩阵的Kronecker积的谱半径的几个不等式.     

矩阵的Kronecker积的性质

利用矩阵初等变换的结果,给出矩阵的Kronecker积的Jordan标准形,进而得到矩阵的Kronecker积的最小多项式和可相似对角化的条件,最后归纳总结了矩阵的Kronecker积的范数和分解的结果.     

带状矩阵Kronecker积结构的矩阵函数与逆矩阵衰减界

带状矩阵以及带状矩阵Kronecker积结构的矩阵函数与逆矩阵的衰减界的研究是近些年非常热门的研究方向,其在数值分析、信号处理等领域有非常重要的应用.主要研究Kronecker积结构形如S_k=kΣi=1I...I︸i-1MI...I︸k-i的矩阵函数与逆矩阵的衰减界.     

投影矩阵的Kronecker积及其相关的正交表

文[3]证明了当P是素数时投影矩阵τp与P水平对称正交表的矩阵象的Kronecker积所包含的正交表是存在的，本文进一步研究了一个非对称正交表的矩阵象和投影矩阵τp的Kronecker积所包含的正交表的存在性，从而推广了文[3]的结论，并且构造出了一些饱和度很高的混合水平正交表．     

矩阵的分块初等变换与分块初等阵及其应用

讨论了矩阵分块初等变换和分块初等阵的定义和性质,利用这一工具研究了行列式的分块运算,分块矩阵的求逆和对称阵的分块合同变换等问题.     

线性混合张量模型及其参数VLS估计

文章将线性混合效应模型(LME)推广至线性混合张量模型.首先建立单变量线性混合张量模型,并推广至多变量线性混合张量模型.利用矩阵向量化方法,结合矩阵函数的导数运算和矩阵的Kronecker积,得出LMEM模型的参数估计特别是方差参数矩阵的估计式,最后给出线性混合张量模型的VLS参数估计式.     

中心对称矩阵的一类问题

利用矩阵的Kronecker积给出了中心对称矩阵的若干特征,并讨论了由特征值和特征向量反构中心对称矩阵的问题.     

关于分块极大极小矩阵性质的研究

利用矩阵分解、矩阵的Hadamard积和数学归纳法研究分块极大极小矩阵的性质.将极大和极小矩阵推广为分块极大和分块极小矩阵.在给出矩阵行列式、逆和特征多项式的同时,得到该类矩阵半正定的充要条件,还讨论了矩阵的无限可分性.     

模糊有限自动机的kronecker积

在矩阵理论框架下,引入了模糊有限自动机转移矩阵,变换矩阵半群以及覆盖概念.定义了模糊有限自动机Kronecker积,讨论了其转移矩阵性质及变换矩阵半群间的覆盖关系.     

矩阵乘法的推广及运用

利用矩阵的Kronecker积定义了一种矩阵乘积"*积",并且对这种乘积的性质进行了研究,发现它对于任意两个矩阵都有意义而且具有通常矩阵乘积的所有性质,并且在一些特殊情况下它比通常的矩阵乘积更和谐对称,而且当在"合适维数"下它就是通常的矩阵乘积,所以可以把这种"*积"看作是对通常矩阵乘积的推广.     

关于r-分块循环矩阵及其对角化问题的探讨

本文给出了r-分块循环矩阵的概念,并利用矩阵的张量积探讨了r-分块循环矩阵的相似类及其对角化问题,得出了一些重要的结论.     

Polynomial numerical hulls of matrix polynomials,II

In this article, we study some algebraic and geometrical properties of polynomial numerical hulls of matrix polynomials and joint polynomial numerical hulls of a finite family of matrices (possibly the coefficients of a matrix polynomial). Also, we study polynomial numerical hulls of basic A-factor block circulant matrices. These are block companion matrices of particular simple monic matrix polynomials. By studying the polynomial numerical hulls of the Kronecker product of two matrices, we characterize the polynomial numerical hulls of unitary basic A-factor block circulant matrices.     

On the inverse of certain patterned sums of matrices with Kronecker product structures

In this article, we derive explicit expressions for the entries of the inverse of a patterned matrix that is a sum of Kronecker products. This matrix keeps the Kronecker structure under matrix inversion, and it is used, for example, in statistics, in particular in the linear mixed model analysis. The obtained results present new and extended existing algorithms for the inversion of the considered patterned matrices. We also obtain a closed-form inverse in terms of block matrices.     

Least Squares Solution of Bivariate Surface Fitting Problems Using Tensor Product Splines

The paper treats bivariate surface fitting problems, where the data points lie on lines parallel to one of the axes. The associated bivariate collocation matrix is investigated as a block Kronecker product of univariate collocation matrices. Based on various properties of this block Kronecker product, such scattered data are characterized where the associated interpolation problem using tensor product splines admits a unique solution.     

Real linear Kronecker product operations

The Kronecker product in the real linear matrix analytic setting is studied. More versatile operations are proposed. Such generalizations are of interest for the same reasons the standard Kronecker product is. To give an example, new preconditioning ideas are suggested. In connection with this, several formulae for the inverse are devised. Orthogonal decompositions of real-entried matrices are derived through introducing new Kronecker product SVDs. Matrix equations are given to illustrate how the Kronecker product structures introduced can arise.     

具有时变区间参数的不确定随机线性系统的均方鲁棒稳定性

研究了一类具有时变区间参数的不确定随机线性系统的均方鲁棒稳定性.利用时变区间矩阵的分解技术、矩阵的Kronecker积的性质和Lyapunov函数法,得到了该系统均方鲁棒稳定的几个充分性条件.通过一个数值例子说明了所得的这些充分性条件的有效性和实用性.     

基于角点处扭矢构造双三次Coons曲面的一种优化方法

本文针对矩形网格角点处的扭矢采用优化方法构造双三次Coons曲面,提出一种新的优化准则来确定角点处的扭矢.首先,通过变分原理,考虑曲面导矢的极小化问题转化的Euler-Lagrange偏微分方程,将该方程应用于每一个Coons块的角点上,引入一个新的极小化问题,其解是Euler-Lagrange偏微分方程的近似最优解.然后,建立一个具有块三对角系数矩阵的线性方程组来求解新的极小化问题.该系数矩阵可以表示为两个相同的形式特殊的矩阵的Kronnecker积,进而可以证明其非奇异性.最后,数值实验验证本文方法的稳定性和有效性.     

Deconvolution and Regularization with Toeplitz Matrices

By deconvolution we mean the solution of a linear first-kind integral equation with a convolution-type kernel, i.e., a kernel that depends only on the difference between the two independent variables. Deconvolution problems are special cases of linear first-kind Fredholm integral equations, whose treatment requires the use of regularization methods. The corresponding computational problem takes the form of structured matrix problem with a Toeplitz or block Toeplitz coefficient matrix. The aim of this paper is to present a tutorial survey of numerical algorithms for the practical treatment of these discretized deconvolution problems, with emphasis on methods that take the special structure of the matrix into account. Wherever possible, analogies to classical DFT-based deconvolution problems are drawn. Among other things, we present direct methods for regularization with Toeplitz matrices, and we show how Toeplitz matrix–vector products are computed by means of FFT, being useful in iterative methods. We also introduce the Kronecker product and show how it is used in the discretization and solution of 2-D deconvolution problems whose variables separate.