Double circulant matrices

Double circulant matrices are introduced and studied. By a matrix-theoretic method, the rank r of a double circulant matrix is computed, and it is shown that any consecutive r rows of the double circulant matrix are linearly independent. As a generalization, multiple circulant matrices are also introduced. Two questions on square double circulant matrices are posed.     

Highly symmetric generalized circulant permutation matrices

In this paper we investigate generalized circulant permutation matrices of composite order. We give a complete characterization of the order and the structure of symmetric generalized k-circulant permutation matrices in terms of circulant and retrocirculant block (0, 1)-matrices in which each block contains exactly one or two entries 1. In particular, we prove that a generalized k-circulant matrix A of composite order n = km is symmetric if and only if either k = m − 1 or k ≡ 0 or k ≡ 1 mod m, and we obtain three basic symmetric generalized k-circulant permutation matrices, from which all others are obtained via permutations of the blocks or by direct sums. Furthermore, we extend the characterization of these matrices to centrosymmetric matrices.     

Some Combinatorial Aspects of Constructing Bipartite-Graph Codes

We propose geometrical methods for constructing square 01-matrices with the same number n of units in every row and column, and such that any two rows of the matrix contain at most one unit in common. These matrices are equivalent to n-regular bipartite graphs without 4-cycles, and therefore can be used for the construction of efficient bipartite-graph codes such that both the classes of its vertices are associated with local constraints. We significantly extend the region of parameters m, n for which there exist an n-regular bipartite graph with 2m vertices and without 4-cycles. In that way we essentially increase the region of lengths and rates of the corresponding bipartite-graph codes. Many new matrices are either circulant or consist of circulant submatrices: this provides code parity-check matrices consisting of circulant submatrices, and hence quasi-cyclic bipartite-graph codes with simple implementation.     

在两循环矩阵类中求解线性方程组反问题的快速算法

本文利用多项式最大公因式 ,给出了线性方程组的反问题在 r-循环矩阵类和对称 r-循环矩阵类中有唯一解的充要条件 ,进而得到线性方程组在 r循环矩阵类和对称 r-循环矩阵类中的反问题求唯一解的算法 .最后给出了应用该算法的数值例子 .     

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.     

Error bounds for linear complementarity problems of DB-matrices

Doubly B-matrices (DB-matrices), which properly contain B-matrices, are introduced by Peña (2003) [2]. In this paper we present error bounds for the linear complementarity problem when the matrix involved is a DB-matrix and a new bound for linear complementarity problem of a B-matrix. The numerical examples show that the bounds are sharp.     

Power Hadamard matrices

We introduce power Hadamard matrices, in order to study the structure of (group) generalized Hadamard matrices, Butson (generalized) Hadamard matrices and other related orthogonal matrices, with which they share certain common characteristics. The new objects turn out to be as interesting, and perhaps as useful, as the objects that motivated them.We develop a basic theory of power Hadamard matrices, explore these relationships, and offer some new insights into old results. For example, we show that all 4×4 Butson Hadamard matrices are equivalent to circulant ones, and how to move between equivalence classes.We provide, among other new things, an infinite family of circulant Butson Hadamard matrices that extends a known class to include one of each positive integer order.Dedication: In 1974 Jennifer Seberry (Wallis) introduced what was then a totally new structure, orthogonal designs, in order to study the existence and construction of Hadamard matrices. They have proved their worth for this purpose, and have also become an object of interest for their own sake and in applications (e.g., [H.J.V. Tarok, A.R. Calderbank, Space-time block codes from orthogonal designs, IEEE Trans. Inf. Theory 45 (1999) 1456-1467. [26]]). Since then many other generalizations of Hadamard matrices have been introduced, including some discussed herein. In the same spirit we introduce a new object showing this kind of promise.Seberry's contributions to this field are not limited to her own work, of which orthogonal designs are but one example—she has mentored many young mathematicians who have expanded her legacy by making their own marks in this field. It is fitting, therefore, that our contribution to this volume is a collaboration between one who has worked in this field for over a decade and an undergraduate student who had just completed his third year of study at the time of the work.     

广义中心对称矩阵平方根的快速算法

本文主要讨论了广义中心对称矩阵平方根的几个性质,并利用这些矩阵的特殊结构,推出了求广义中心对称矩阵平方根的快速算法。与传统算法相比,我们的结构算法无论在内存,还是在计算量方面,都有相当的节省。     

On nonsingular regular magic squares of odd order

Using centroskew matrices, we provide a necessary and sufficient condition for a regular magic square to be nonsingular. Using latin squares and circulant matrices we describe a method of construction of nonsingular regular magic squares of order n where n is an odd prime power.     

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.     

A note on certain matrices with h(x) – Fibonacci quaternion polynomials

In this paper we consider a g – circulant, right circulant, left circulant and a special kind of a tridiagonal matrices whose entries are h(x) – Fibonacci quaternion polynomials. We present the determinant of these matrices and with the tridiagonal matrices we show that the determinant is equal to the nth term of the h(x) – Fibonacci quaternion polynomial sequences.     

循环矩阵本原指数的上界估计

本文给出了循环矩阵本原指数上界的新的估计及一种由级数较低的循环矩阵的本原指数估计级数较高的循环矩阵的本原指数的方法，解决了一类循环矩阵本原指数的计算问题．     

Residual methods for the large-scale matrix pth root and some related problems

The problem of finding the pth root of a matrix has received special attention in the last few years. Standard approaches for this problem include and combine some variations of Newton’s method, which in turn involve matrix factorizations that, in general, are not suitable for large-scale problems. Motivated by some recently developed low-cost iterative schemes for nonlinear problems, we consider and analyze specialized residual methods that only require a few matrix-matrix products per iteration, and hence are suitable for the large-scale case. As a by-product we also discuss the advantages of residual methods for general nonlinear problems whose variables separate. Preliminary and encouraging numerical results are presented for computing pth roots of large-scale symmetric and positive definite matrices, for different values of p.     

Algorithms for Finding the Inverses of Factor Block Circulant Matrices

In this paper,algorithms for finding the inverse of a factor block circulant matrix, a factor block retrocirculant matrix and partitioned matrix with factor block circulant blocks over the complex field are presented respectively.In addition,two algorithms for the inverse of a factor block circulant matrix over the quaternion division algebra are proposed.     

On Eisenbud's and Wigner's R-matrix: A general approach

The main objective of this paper is to give a rigorous treatment of Wigner's and Eisenbud's R-matrix method for scattering matrices of scattering systems consisting of two selfadjoint extensions of the same symmetric operator with finite deficiency indices. In the framework of boundary triplets and associated Weyl functions an abstract generalization of the R-matrix method is developed and the results are applied to Schrödinger operators on the real axis.     

The nonidealness index of rank-ideal matrices

A polyhedron is integral if all its extreme points have 0, 1 components and in this case the matrix M is called ideal. When Q has fractional extreme points, there are different ways of classifying how far M is away from being ideal, through the polyhedral structure of Q. In this sense, Argiroffo, Bianchi and Nasini (2006) [1] defined a nonidealness index analogous to an imperfection index due to Gerke and McDiarmid (2001) [10].In this work we determine the nonidealness index of rank-ideal matrices (introduced by the authors (2008)). It is known that evaluating this index is NP-hard for any matrix. We provide a tractable way of evaluating it for most circulant matrices, whose blockers are a particular class of rank-ideal matrices, thereby following similar lines as done for the imperfection ratio of webs due to Coulonges, Pêcher and Wagler (2009) [7].Finally, exploiting the properties of this nonidealness index, we identify the facets of the set covering polyhedron of circulant matrices, having maximum strength with respect to the linear relaxation, according to a measure defined by Goemans (1995) [9].     

对角因子分块循环矩阵的对角化和谱分解

导出了对角因子分块循环矩阵的概念，把循环矩阵的对角化和谱分解推广到具有对角因子循环结构的分块矩阵中去．     

Circulant Wavelet Preconditioners for Solving Elliptic Differential Equations and Boundary Integral Equations

In this paper is discussed solving an elliptic equation and a boundary integral equation of the second kind by representation of compactly supported wavelets. By using wavelet bases and the Galerkin method for these equations, we obtain a stiff sparse matrix that can be ill-conditioned. Therefore, we have to introduce an operator which maps every sparse matrix to a circulant sparse matrix. This class of circulant matrices is a class of preconditioners in a Banach space. Based on having some properties in the spectral theory for this class of matrices, we conclude that the circulant matrices are a good class of preconditioners for solving these equations. We called them circulant wavelet preconditioners (CWP). Therefore, a class of algorithms is introduced for rapid numerical application.     

具有位数3和4的双随机循环矩阵中的素元

本文研究了双随机循环矩阵中素元的分类问题．由于任一n阶双随机循环矩阵都可以唯一地表示为移位的n-1次一元多项式,从而可把双随机循环矩阵中素元的分类问题简化为解双随机循环矩阵上的一个方程．应用此原理,本文完全解决了判别具有位数3的n阶双随机循环矩阵是否为素元的问题,并给出了n阶双随机循环矩阵中一类具有位数4的素元．