广义模糊幂零矩阵

路代数是加法幂等的半环,它包括了布尔代数,模糊代数,分配格及斜坡.因此布尔矩阵,模糊矩阵,格矩阵及斜矩阵都是路代数上的典型矩阵.广义模糊幂零矩阵指的就是路代数上的幂零矩阵.在2010年,Tan研究了路代数上矩阵的幂零性.在Tan的基础上继续讨论了路代数上幂零矩阵的幂零指数.     

一类极小谱任意符号模式矩阵

研究了一类恰含2n个非零元的n(n≥5)阶零-非零模式矩阵P.证明了将P中所有非零元规定适当的符号,或换为适当的复数,分别可得到一个极小谱任意符号模式矩阵A和一个极小谱任意的复符号模式矩阵S.     

3-幂零矩阵的相似等价类的计数

主要对定义在一般数域上的3-幂零矩阵的相似等价类的个数问题进行探讨.从中得出n阶3-幂零矩阵秩的范围、n阶3-幂零矩阵的相似等价类的个数的计算公式,以及秩为r的所有n阶3-幂零矩阵的相似等价类的个数的计算公式.     

一类幂零矩阵幂零指标的特征

设(L, ,·)是一个incline. 本文给出了一个incline上幂零矩阵幂零指标的特征.其结果改进了文[4]中的相应结论.     

幂等右侧Quantale上的幂零矩阵

讨论幂等右侧Quantale上的幂零矩阵的若干性质，给出了幂等右侧Quantale上的矩阵为幂零矩阵的充要条件，得到了幂零矩阵的幂零指数的刻画定理。     

一类新的谱任意复符号模式

设S是一个n阶复符号模式.如果对每一个首项系数为1的n次复系数多项式f(λ),都存在一个复矩阵C∈Q(S),使C的特征多项式就是f(λ),则称S是一个谱任意复符号模式.也可以这样描述:如果n阶复符号模式S的谱集是由所有n阶复矩阵的谱构成的集合,则称S是谱任意复符号模式.本文利用Schur补和行化简方法,给出了一类仅含3n个非零元的谱任意复符号模式.     

关于幂零模糊矩阵的若干结果

本文得出模糊矩阵幂零的几个充要条件,并讨论了幂零模糊矩阵与强可迁矩阵的关系。     

幂零Fuzzy矩阵的一些结论及幂零度的求法

本文给出计算幂零Fuzzy矩阵的幂零度的两个办法,同时给出一个有有向Hamilton路的有向图的一个充分条件.     

一道高等代数常见习题的自然延伸

分析了一道高等代数常见习题的自然延伸,给出了幂零矩阵的幂零指数的上界估计.     

关于广义Dedekind群

如果群G的任意循环子群H满足|HG:H|≤p,其中p是素数,那么称G是C*(p)-群.若群G是有限C*(p)-p-群,则当p＞3时,该群的幂零类至多为2;若p=3,该群的幂零类至多为3,而且当cl(G)=3时,exp(G)=9;同时,若G与任意有限C*(p)-p群G × K直积是C*(p)-P-群G×K,则G是初等阿贝尔p-群.最后还对局部幂零的C*(p)-群进行了探讨.     

谱任意的符号模式矩阵

一个n阶符号模式矩阵A称为是谱任意的,如果对任意的实系数n次首1多项式r(x),在A的定性矩阵类Q(A)中至少存在一个实矩阵B,使得B的特征多项式是r(x),文中证明了当n为奇数时n阶谱任意符号模式矩阵是存在的。     

一类新的极小谱任意符号模式

若给定任意一个$n$次首一实系数多项式$f(\lambda)$,都存在一个实矩阵$B\in Q(A)$, 使得$B$的特征多项式为$f(\lambda)$,则称$A$为谱任意符号模式. 如果一个谱任意符号模式的任意非零元被零取代后所得到的符号模式不是谱任意,那么这个谱任意符号模式称为极小谱任意符号模式.本文证明一类极小谱任意符号模式.     

Allow problems concerning spectral properties of sign pattern matrices: A survey

An n×n sign pattern matrix has entries in {+,-,0}. This paper surveys the following problems concerning spectral properties of sign pattern matrices: sign patterns that allow all possible spectra (spectrally arbitrary sign patterns); sign patterns that allow all inertias (inertially arbitrary sign patterns); sign patterns that allow nilpotency (potentially nilpotent sign patterns); and sign patterns that allow stability (potentially stable sign patterns). Relationships between these four classes of sign patterns are given, and several open problems are identified.     

幂零--中心化方法

An nxn complex sign pattern(ray pattern) S is said to be spectrally arbitrary if for every monic nth degree polynomial f(λ) with coefficients from C,there is a complex matrix in the complex sign pattern class(ray pattern class) of S such that its characteristic polynomial is f(λ).We derive the Nilpotent-Centralizer methods for spectrally arbitrary complex sign patterns and ray patterns,respectively.We find that the Nilpotent-Centralizer methods for three kinds of patterns(sign pattern,complex sign pattern,ray pattern) are the same in form.     

On reducible matrix patterns

A tool to study the inertias of reducible nonzero (resp. sign) patterns is presented. Sumsets are used to obtain a list of inertias attainable by the pattern 𝒜 ⊕ ? dependent upon inertias attainable by patterns 𝒜 and ?. It is shown that if ? is a pattern of order n, and 𝒜 is an inertially arbitrary pattern of order at least 2(n ? 1), then 𝒜 ⊕ ? is inertially arbitrary if and only if ? allows the inertias (0, 0, n), (0, n, 0) and (n, 0, 0). We illustrate how to construct other reducible inertially (resp. spectrally) arbitrary patterns from an inertially (resp. spectrally) arbitrary pattern 𝒜 ⊕ ?, by replacing 𝒜 with an inertially (resp. spectrally) arbitrary pattern 𝒮. We identify reducible inertially (resp. spectrally) arbitrary patterns of the smallest orders that contain some irreducible components that are not inertially (resp. spectrally) arbitrary. It is shown there exist nonzero (resp. sign) patterns 𝒜 and ? of orders 4 and 5 (resp. 4 and 4) such that both 𝒜 and ? are non-inertially-arbitrary, and 𝒜 ⊕ ? is inertially arbitrary.     

Several spectrally arbitrary ray patterns

A ray pattern A of order n is said to be spectrally arbitrary if given any monic nth degree polynomial f(x) with coefficients from ?, there exists a matrix realization of A such that its characteristic polynomial is f(x). An n?×?n ray pattern A is said to be minimally spectrally arbitrary if replacing any nonzero entry of A by zero destroys this property. In this article, several families of ray patterns are presented and proved to be minimally spectrally arbitrary. We also show that for n?≥?5, when A _n is spectrally arbitrary, then it is minimally spectrally arbitrary.     

k次幂非正的符号模式矩阵

本首先对使得A^K≤0的符号模式矩阵A进行了刻画（k为任意正整数），进而决定了这类矩阵中负元个数的最大值。最后给出了使得A^2≤0的符号模式矩阵A的充分必要条件。     

A CLASS OF SPECTRALLY ARBITRARY RAY PATTERNS

An n × n ray pattern A is said to be spectrally arbitrary if for every monic nth degree polynomial f(x) with coefficients from C, there is a complex matrix in the ray pattern class of A such that its characteristic polynomial is f(x). In this paper, a family ray patterns is proved to be spectrally arbitrary by using Nilpotent-Jacobian method.     

Spectrally arbitrary ray patterns

An n×n ray pattern A is said to be spectrally arbitrary if for every monic nth degree polynomial f(x) with coefficients from C, there is a matrix in the pattern class of A such that its characteristic polynomial is f(x). In this article the authors extend the nilpotent-Jacobi method for sign patterns to ray patterns, establishing a means to show that an irreducible ray pattern and all its superpatterns are spectrally arbitrary. They use this method to establish that a particular family of n×n irreducible ray patterns with exactly 3n nonzeros is spectrally arbitrary. They then show that every n×n irreducible, spectrally arbitrary ray pattern has at least 3n-1 nonzeros.