几种矩阵逆的连续性

非奇异矩阵的逆是矩阵元素的连续函数．学者们也对矩阵广义逆的连续性有所研究．本文应用矩阵分裂和两个矩阵之和的逆的展开式,给出了一般非奇异矩阵,M-矩阵和H-矩阵的逆的连续性．当一些合理的条件满足时,这几种矩阵的逆是连续的．     

Full spark frames and totally positive matrices

In this paper we establish a connection between full spark frames and totally nonsingular matrices. Then we provide a method for constructing infinite totally positive matrices which make up a subclass of the class of totally nonsingular matrices. Using this method we then construct a family of infinite totally positive matrices parameterized by non-negative numbers which contains, as the simplest case, the infinite Pascal matrix. The paper ends with some examples and comments on full spark frames.     

Tridiagonal and upper triangular inverse M-matrices

In this paper we characterize the nonnegative nonsingular tridiagonal matrices belonging to the class of inverse M-matrices. We give a geometric equivalence for a nonnegative nonsingular upper triangular matrix to be in this class. This equivalence is extended to include some reducible matrices.     

M-matrix characterizations.I—nonsingular M-matrices

The purpose of this survey is to classify systematically a widely ranging list of characterizations of nonsingular M-matrices from the economics and mathematics literatures. These characterizations are grouped together in terms of their relations to the properties of (1) positivity of principal minors, (2) inverse-positivity and splittings, (3) stability and (4) semipositivity and diagonal dominance. A list of forty equivalent conditions is given for a square matrix A with nonpositive off-diagonal entries to be a nonsingular M-matrix. These conditions are grouped into classes in order to identify those that are equivalent for arbitrary real matrices A. In addition, other remarks relating nonsingular M-matrices to certain complex matrices are made, and the recent literature on these general topics is surveyed.     

Characterization of ray pattern matrix whose determinantal region has two components after deleting the origin

Ray nonsingular matrices are generalizations of sign nonsingular matrices. The problem of characterizing ray nonsingular matrices is still open. The study of the determinantal regions R_A of ray pattern matrices is closely related to the study of ray nonsingular matrices. It was proved that if R_A?{0} is disconnected, then it is a union of two opposite open sectors (or open rays). In this paper, we characterize those ray patterns whose determinantal regions become disconnected after deleting the origin. The characterization is based on three classes (F1), (F2) and (F3) of matrices, which can further be characterized in terms of the sets of the distinct signed transversal products of their ray patterns. Moreover, we show that in the fully indecomposable case, a matrix A is in the class (F1) (or (F2), respectively) if and only if A is ray permutation equivalent to a real SNS (or non-SNS, respectively) matrix.     

Intervals of totally nonnegative matrices

Totally nonnegative matrices, i.e., matrices having all their minors nonnegative, and matrix intervals with respect to the checkerboard ordering are considered. It is proven that if the two bound matrices of such a matrix interval are nonsingular and totally nonnegative (and in addition all their zero minors are identical) then all matrices from this interval are also nonsingular and totally nonnegative (with identical zero minors).     

To the problem of multiplicativity of canonical diagonal forms of matrices over the domain of principal ideals

We study the structure of nonsingular matrices over the domain of principal ideals that possess the property of multiplicativity of canonical diagonal forms. In particular, we establish necessary and sufficient conditions of multiplicativity of canonical diagonal forms of nonsingular matrices over this domain.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 11, pp. 1581–1584, November, 1995.     

Ostrowski定理的推广与非奇异H-矩阵的实用判定

利用α2-双对角占优理论,给出了几个判定非奇异H-矩阵的充分条件,扩大了非奇异H-矩阵的判定范围,并给出了相应的数值算例说明结果的有效性．     

Practical criteria for H-matrices

In this paper, a set of criteria of nonsingular H-matrices are discussed. The paper introduces the concept of α-bidiagonally dominant matrices and gives an equivalent condition of strictly α-bidiagonally dominant matrices. According to the given condition, some new practical criteria of nonsingular H-matrices are obtained. Finally, some numerical examples are given.     

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.     

H-矩阵的实用判定及谱分布

1引言及记号因为非奇异H-矩阵主对角元非零,所以本文总假定所涉及矩阵主对角元非零,并且设A=(aij)∈Cn×n为n阶复方阵,N={1,2,…,n}．记N1={i∈N |Pi(A)<|aii|Pi(A)}, N4={i∈N | |aii|≥Pi(A)>Ri(A)}, N5={i∈N | |aii|>Pi(A)=Ri(A)},N0={i∈N | |aii|≤Ri(A),|aii|≤Pi(A)},即N=N1∪N2∪N3∪N4∪N5∪N0．     

Monotone positive stable matrices

The class of real matrices which are both monotone (inverse positive) and positive stable is investigated. Such matrices, called N-matrices, have the well-known class of nonsingular M-matrices as a proper subset. Relationships between the classes of N-matrices, M-matrices, nonsingular totally nonnegative matrices, and oscillatory matrices are developed. Conditions are given for some classes of matrices, including tridiagonal and some Toeplitz matrices, to be N-matrices.     

Taussky定理推广与应用

本文在Ｃａｓｓｉｎｉ卵形域上推广了Ｔａｕｓｓｋｙ定理。所得结果修正了Ｂｒａｕｅｒ定理,作为应用给出不可约双对角占优矩阵非奇异的充要条件,最后把基本结果推广到分块矩阵上。     

MP matrices

MP matrices are those real matrices which possess a nonnegative, nonsingular l-inverse. The difficult task of describing the class of MP matrices is investigated.     

On parametrization of totally nonpositive matrices and applications

The problem of accurate computations for totally non‐negative matrices has been studied; however, it remains open for other sign regular matrices. One major obstacle is that there is no known parametrization of these matrices. The main contribution of the present work is that we provide such parametrization of nonsingular totally nonpositive matrices. A useful application of our results is that these parameters can determine accurately the entries of the inverse of a nonsingular totally nonpositive matrix. Copyright © 2011 John Wiley & Sons, Ltd.     

Accurate computations of matrices with bidiagonal decomposition using methods for totally positive matrices

A class of sign‐symmetric P‐matrices including all nonsingular totally positive matrices and their inverses as well as tridiagonal nonsingular H‐matrices is presented and analyzed. These matrices present a bidiagonal decomposition that can be used to obtain algorithms to compute with high relative accuracy their singular values, eigenvalues, inverses, or their LDU factorization. Copyright © 2012 John Wiley & Sons, Ltd.     

The conditioning of Toeplitz band matrices

In this paper, practical conditions to check the well-conditioning of a family of nonsingular Toeplitz band matrices are obtained. All the results are based on the location of the zeros of a polynomial associated with the given family of Toeplitz matrices.The same analysis is also used to derive uniform componentwise bounds for the entries of the inverse matrices in such family.     

一类Toeplitz矩阵的群逆

Ｔｏｅｐｌｉｔｚ矩阵Ｔｎ＝（ｔｉ－ｊ）ｎｉ，ｊ＝０在信号处理、系统理论、逼近论、正交多项式、积分方程数值解等许多领域常常遇到．易知，Ｔｏｅｐｌｉｔｚ矩阵Ｔｎ的逆矩阵一般不再是Ｔｏｅｐｌｉｔｚ矩阵．１９７２年，Ｇｏｈｂｅｒｇ和Ｓｅｍｅｎｃｕｌ［１］给出...     

分支数≤7时S^2NS有向图的禁用子图刻划

实方阵A称为强符号非异阵（S~2NS阵）,若任一与A符号模式相同的矩阵非异且其逆的符号模式也一致。若一个有向图是某一S~2NS阵对应赋号有向图的基础有向图,称为S~2NS有向图。本文用禁用子图形式给出了分支数≤7时有向图成为S~2NS有向图的刻划,同时部分地解决了[2]和[4]中提出的问题。     

基于严格α-双链对角占优矩阵的非奇H-矩阵的判定准则

研究了非奇H-矩阵的判定问题.先给出了几个判定严格α-双链对角占优矩阵的充要条件,进一步利用矩阵对角占优理论得到了判定非奇H-矩阵的一些充分条件,推广和改进了已有的相关结果,并用数值算例说明了这些判定方法的有效性.