友循环矩阵求逆和广义逆的Euclid算法

利用多项式的Euclid算法给出了任意域上非奇异的友循环矩阵求逆矩阵的一个新算法,该算法同时推广到用于求任意域上奇异友循环矩阵的群逆和Moore-Penrose逆,最后给出了应用该算法的数值例子。     

求鳞状因子循环矩阵的逆阵及广义逆阵的快速算法

利用多项式快速算法，给出了求鳞状因子循环矩阵的逆阵、自反g-逆、群逆及Moore-Penrose逆的快速算法。该算法避免了一般快速算法中，要计算大量的三角函数等可能带来误差及影响效率的问题。该算法仅用到鳞状因子循环矩阵的第一行元素及对角阵D中的常数d1，d2，…，dn进行计算，在计算机上实现时只有舍入误差。特别地，在有理数域上用计算机求得的结果是精确的。     

鳞状因子循环矩阵方程解的条件与求解的快速算法

利用多项式快速算法,给出了鳞状因子循环矩阵方程AX=b可解的条件与求解的快速算法.当鳞状因子循环矩阵非奇异时,该快速算法求出线性方程组的唯一解;当鳞状因子循环矩阵奇异时,该快速算法求出线性方程组的特解与通解.该快速算法仅用到鳞状因子循环矩阵的第一行元素及对角矩阵中的对角上的常数进行计算.在计算机上实现时只有舍入误差.特别地,在有理数域上用计算机求得的结果是精确的.     

Cline分块矩阵的极小范数广义逆

矩阵广义逆的递推算法在统计推断、模式识别以及解析动力学等领域有着广泛的应用背景.本文利用范数极小化方法给出了计算Cline分块矩阵的极小范数广义逆及逆的两个一般递推表示式,推广了已有的结果.     

多项式正交滤波器与共轭正交滤波器组

主要讨论了多项式正交滤波器和共轭正交滤波器组的构造方法，首先利用Riesz引理和特殊的余弦三角形多项式，给出了一种多项式正交滤波器的构造算法，该算法可以构造出一系列特性各异的紧支撑正交小波基；还给出了由一个矩阵CQFs派生多个新的矩阵CQFs的共轭正交滤波器组算法,包括由低阶矩阵CQFs构造高阶矩阵CQFs。     

多项式矩阵带余除法的总结与应用

通过查阅文献了解到一元、二元及多元多项式带余除法都有着相同的思想,我们不由的会想到这一思想是否也可以应用在矩阵中．答案是肯定的,原因是多项式的系数又可构成矩阵的形式,但是在做多项式矩阵带余除法时稍有差异,也就是矩阵之间不可交换的定律影响着带余除法的求解过程．由于带余除法除了在多项式整除中占主导位置,也普遍应用干生活中,再加上以矩阵为系数的多项式是一个应用很广泛的方法,因此,对多项式矩阵的带余除法的总结和研究是很有必要的．所以本文首先介绍多项式矩阵涉及到的一些概念和结论,然后引进一元多项式矩阵的带余除法,之后给出其应用,接着,在此基础上结合它的带余除法思想,给出多元多项式矩阵带余除法的一个应用．同时也罗列了多项式矩阵带余除法的相关知识,收集了主题为多项式矩阵带余除法应用的文献,对其整理并做文献综述．     

g-r循环矩阵求逆的快速算法

利用FFT,给出g-r循环矩阵求逆的快速算法,计算复杂性为O(n log2 n) (g 2)n。     

部分逆M矩阵2-弦图的完备问题

本文采用图论的方法对任意阶部分逆M矩阵,当其对应的图为2-弦图时,研究了其逆M矩阵的完备问题。给出了完备定理以及具体完备的算法。     

广义Pascal矩阵及其代数性质

Pascal矩阵及其推广形式的代数性质的研究在电子工程、组合数学、快速算法、微分方程数值解等领域有着广泛的应用。本文利用多项式空间基变换的方法,新给出了几类广义Pascal矩阵,即广义左-Pascal矩阵、广义右-Pascal矩阵和推广的广义Pascal矩阵的一些代数性质的简洁证明,同时给出了这几类广义Pascal矩阵一些新的代数性质。     

Loewner方程组极小范数最小二乘解的快速算法

本文给出了求以m×n阶Loewner矩阵为系数阵的线性方程组极小范数最小二乘解的快速算法。     

分配伪格上的可逆矩阵(英文)

本文介绍了分配伪格上的可逆矩阵,获得分配伪格上矩阵可逆的一些充分必要条件,也证明了在整的分配伪格上,矩阵可逆当且仅当它是一个置换矩阵.这些结果推广了分配格和交换坡上已有关于可逆矩阵的结果.     

Block circulant matrices and applications in free vibration analysis of cyclically repetitive structures

In this paper, block circulant matrices and their properties are investigated. Basic concepts and the necessary theorems are presented and then their applications are discussed. It is shown that a circulant matrix can be considered as the sum of Kronecker products in which the first components have the commutativity property with respect to multiplication. The important fact is that the method for block diagonalization of these matrices is much simpler than the previously developed methods, and one does not need to find an additional matrix for orthogonalization. As it will be shown not only the matrices corresponding to domes in the form of Cartesian product, strong Cartesian product and direct product are circulant, but for other structures such as diamatic domes, pyramid domes, flat double layer grids, and some family of transmission towers these matrices are also block circulant.     

Analysis of structures transformable to circulant form using U-transformation and Kronecker products

In this paper, eigenvalues and eigenvectors of the specific types of structural matrices are studied, and a simple method is presented for calculating their eigenvalues. First, the required formulation to diagonalize circulant and block circulant matrices is presented by using U-matrix transformation. Then utilizing the method of this paper, matrices with non-circulant forms are converted into their circulant counterpart matrices. In order to demonstrate the efficiency of the method, some examples are provided using numerical methods such as finite differences, finite element and finite stripe methods.     

Some results on permutation polynomials over finite fields

This paper deals with two questions concerning permutation polynomials in several variables. Lidl and Niederreiter have considered the problem of when a sum of permutation polynomials in disjoint sets of variables is itself a permutation polynomial, and in the case of prime fields have shown that it is necessary and sufficient that at least one summand be a permutation polynomial. They also showed that in the case of non-prime fields this condition is not necessary. In this paper, a necessary and sufficient condition is obtained for the general case which specialises to the previous result for prime fields. The second part extends a criterion of Niederreiter for permutation polynomials over prime fields to any finite field.     

Newton-Shamanskii Method for a Quadratic Matrix Equation Arising in Quasi-Birth-Death Problems

In order to determine the stationary distribution for discrete time quasi-birth-death Markov chains, it is necessary to find the minimal nonnegative solution of a quadratic matrix equation. The Newton-Shamanskii method is applied to solve this equation, and the sequence of matrices produced is monotonically increasing and converges to its minimal nonnegative solution. Numerical results illustrate the effectiveness of this procedure.     

求解四元数矩阵方程AHXA=B的基于矩阵半张量积的新方法

四元数线性系统在控制理论和工程中有广泛的应用。利用矩阵半张量积对四元数矩阵方程进行研究，提出四元数矩阵的一种实向量表示并研究其性质。结合实向量表示与矩阵半张量积，给出四元数矩阵方程A^HXA=B的极小范数Hermitian解的存在条件及通解表达式，并且给出相应算法。数值实验证明了实向量表示方法的可行性。     

缺项算子矩阵的二阶代数补Ⅱ

对于任意给定的二阶多项式p(t)以及缺项算子矩阵（AC?B)，本文给出存在补矩阵T＝（^AC XB)使得p(T)=0的充分必要条件，而且给出这种二阶代数补的完全刻画。     

Exact and approximate algorithms for part-machine clustering based on a relationship between interval graphs and Robinson matrices

An important manufacturing cell formation problem requires permutations of the rows (parts) and columns (machines) of a part-machine incidence matrix such that the reordered matrix exhibits a block-diagonal form. Numerous objective criteria and algorithms have been proposed for this problem. In this paper, a new perspective is offered that is based on the relationship between the consecutive ones property associated with interval graphs and Robinson structure within symmetric matrices. This perspective enables the cell formation problem to be decomposed into two permutation subproblems (one for rows and one for columns) that can be solved optimally using dynamic programming or a branch-and-bound algorithm for matrices of nontrivial size. A simulated annealing heuristic is offered for larger problem instances. Results pertaining to the application of the proposed methods for a number of problems from the literature are presented.