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.     

本质(m,l)幂等矩阵的特征研究

若有最小正整数m使当m>l时A(m)=A(l)成立,称A为本质(m,l)幕等矩阵.本文讨论了本质(m,l)幕等矩阵的特征.作为应用,给出了本质m对合、本质m幂等矩阵的等价刻画,讨论了最小多项式与本质(m,l)幕等矩阵的一些关系.     

广义m-幂矩阵的等价条件

利用矩阵的秩和齐次线性方程组解空间的维数,给出了广义m-幂矩阵的5个等价条件,推广了幂幺矩阵和m次幂等矩阵的相应结论.此外,把广义m-幂矩阵的这几个等价条件推广到了广义m-幂变换中.     

m次幂等矩阵的等价条件

利用矩阵的秩和齐次线性方程组解空间的维数,给出了m(m≥2)次幂等矩阵的一些等价条件,推广了2,3次幂等矩阵的相应结果.此外,所获结果还给推广到了m次幂等线性变换中.     

广义(u,v)幂等矩阵

首先定义了一类新的矩阵一广义(u,v)幂等矩阵,然后研究了它的等价刻画,从而推广了(u,v)幂等矩阵、m幂幺矩阵、m幂等矩阵的一些相应结果.此外,也探讨了广义(u,v)幂等矩阵的性质,以及广义(u,v)幂等矩阵与广义m幂矩阵的关系.     

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.     

幂保持加法映射

本文得到了域上n×n矩阵代数的幂保持加法映射的刻画,同时刻画了对称矩阵的幂保持加法映射.     

幂等矩阵的性质及其推广

首先对幂等矩阵的简单性质进行了归纳总结,接着论证了幂等矩阵的等价条件及其特征值的取值范围,并讨论了幂等矩阵与实对称矩阵的关系、幂等矩阵与其伴随矩阵的特征值和特征向量的对应关系及幂等矩阵在群逆中的一个性质.最后讨论了幂等矩阵的两种分解形式.     

关于2×2矩阵代数保立方幂等的单射

刻画了2×2全矩阵代数保立方幂等映射,利用保立方幂等映射象与原象矩阵对应元素相等这个特点计算出相应的结果,得出特征不等于2,3,5域上2×2全矩阵代数保立方幂等单射的一个具体的刻画.     

m次数量幂等矩阵线性组合的可逆性

如果有非零数λ与μ使Pm=λP,Qm=μQ,则称P,Q分别是由λ,μ确定的m次数量幂等矩阵．本文证明了,若有非零数a与b,当μam-1（-1）m-1μbm-1≠0时,使可交换的分别由λ,μ确定的m次数量幂等矩阵P,Q的线性组合aP＋bQ是可逆的,那么对任意非零数u,u,当λμm-1-（-1）m-1μvm-1≠0时,uP＋vQ也是可逆的．本文主要结果和方法的应用,可以推广已有文献的2次、3次幂等矩阵的线性组合可逆的结论．     

Dn中幂等元的一种关系

本研究了Dn中幂等元的某些性质。给出了幂等元的另一个等价刻划以及两幂等元之积仍是幂等元的一个充要条件。     

Inequalities for Gondran-Minoux rank and idempotent semirings

The rank-sum, rank-product, and rank-union inequalities for Gondran-Minoux rank of matrices over idempotent semirings are considered. We prove these inequalities for matrices over quasi-selective semirings without zero divisors, which include matrices over the max-plus semiring. Moreover, it is shown that the inequalities provide the linear algebraic characterization for the class of quasi-selective semirings. Namely, it is proven that the inequalities hold for matrices over an idempotent semiring S without zero divisors if and only if S is quasi-selective. For any idempotent semiring which is not quasi-selective it is shown that the rank-sum, rank-product, and rank-union inequalities do not hold in general. Also, we provide an example of a selective semiring with zero divisors such that the rank-sum, rank-product, and rank-union inequalities do not hold in general.     

Products of idempotent matrices over integral domains

We say that a ring R has the idempotent matrices property if every square singular matrix over R is a product of idempotent matrices. It is known that every field, and more generally, every Euclidean domain has the idempotent matrices property. In this paper we show that not every integral domain has the idempotent matrices property and that if a projective free ring has the idempotent matrices property then it must be a Bezout domain. We also show that a principal ideal domain has the idempotent matrices property if and only if every fraction a/b with b≠0 has a finite continued fraction expansion. New proofs are also provided for the results that every field and every Euclidean domain have the idempotent matrices property.     

Matrices over a commutative ring as sums of three idempotents or three involutions

Motivated by Hirano-Tominaga’s work on rings for which every element is a sum of two idempotents and by de Seguins Pazzis’s results on decomposing every matrix over a field of positive characteristic as a sum of idempotent matrices, we address decomposing every matrix over a commutative ring as a sum of three idempotent matrices and, respectively, as a sum of three involutive matrices.     

Idempotent matrix lattices over distributive lattices

In this paper, the partially ordered set of idempotent matrices over distributive lattices with the partial order induced by a set of lattice matrices is studied. It is proved that this set is a lattice; the formulas for meet and join calculation are obtained. In the lattice of idempotent matrices over a finite distributive lattice, all atoms and coatoms are described. We prove that the lattice of quasi-orders over an n-element set Qord(n) is not graduated for n ≥ 3 and calculate the greatest and least lengths of maximal chains in this lattice. We also prove that the interval ([I, J]_≤, ≤) of idempotent (n × n)-matrices over {ie879-01}-lattices is isomorphic to the lattice of quasi-orders Qord(n). Using this isomorphism, we calculate the lattice height of idempotent {ie879-02}-matrices. We obtain a structural criterion of idempotent matrices over distributive lattices. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 4, pp. 121–144, 2007.     

Products of idempotent matrices

For some years it has been known that every singular square matrix over an arbitrary field F is a product of idempotent matrices over F. This paper quantifies that result to some extent. Main result: for every field F and every pair (n,k) of positive integers, an n×n matrix S over F is a product of k idempotent matrices over F iff rank(I ? S)?k· nullity S. The proof of the “if” part involves only elementary matrix operations and may thus be regarded as constructive. Corollary: (for every field F and every positive integer n) each singular n×n matrix over F is a product of n idempotent matrices over F, and there is a singular n×n matrix over F which is not a product of n ? 1 idempotent matrices.     

Invertible Matrices over Finite Additively Idempotent Semirings

We investigate invertible matrices over finite additively idempotent semirings. The main result provides a criterion for the invertibility of such matrices. We also give a construction of the inverse matrix and a formula for the number of invertible matrices.     

The complete equivalence canonical form of four matrices over an arbitrary division ring

In this paper, we give the complete structures of the equivalence canonical form of four matrices over an arbitrary division ring. As applications, we derive some practical necessary and sufficient conditions for the solvability to some systems of generalized Sylvester matrix equations using the ranks of their coefficient matrices. The results of this paper are new and available over the real number field, the complex number field, and the quaternion algebra.     

Idempotent factorization of matrices

Necessary and sufficient conditions are presented for a square matrix over an arbitrary field to be a product of k ≥ 1 idempotent matrices of prescribed nullities.