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The Study of Root Subspace Decomposition between characteristic polynomials and Minimum polynomial

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Open Journal of Applied Sciences 2024年第7期14卷 1637-1647页

作者： Lilong Kang Yu Wang Yingling Liu College of Mathematics and Statistics Sichuan University of Science and Engineering Zigong China

Let Abe the linear transformation on the linear space V in the field P, Vλibe the root subspace corresponding to the characteristic polynomial of the eigenvalue λi, and Wλibe the root subspace corresponding to the minimum polynomial of λi. Consider the problem of whether Vλiand Wλiare equal under the condition that the characteristic polynomial of Ahas the same eigenvalue as the minimum polynomial (see Theorem 1, 2). This article uses the method of mutual inclusion to prove that Vλi=Wλi. Compared to previous studies and proofs, the results of this research can be directly cited in related works. For instance, they can be directly cited in Daoji Meng’s book “Introduction to Differential Geometry.”

关键词： characteristic polynomial Minimum polynomial Root Subspace

Rank-generating functions and Poincaré polynomials of lattices in finite orthogonal space of even characteristic

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Frontiers of Mathematics in China 2024年第4期19卷 181-189页

作者： Feng XU Yanbing ZHAO Yuanji HUO Office of Academic Affairs Xuanhua Vocational College of Science and TechnologyZhangjiakou 075100China Department of Basic Courses Zhangjiakou Vocational and Technical CollegeZhangjiakou 075051China Department of Mathematics Hebei North UniversityZhangjiakou 075000China

The geometry of classical groups over finite fields is widely used in many *** this paper,we study the rank-generating function,the characteristic polynomial,and the Poincarépolynomial of lattices generated by the or... 详细信息

关键词： Lattices orthogonal space of even characteristic rank-generating function characteristic polynomial Poincarépolynomial

Lattices Generated by Joins of the Flats in Orbits under Finite Affine-singular Symplectic Group and its characteristic polynomials

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Acta Mathematicae Applicatae Sinica 2017年第4期33卷 919-932页

作者： You GAO Yan-yan XUE Yu-ting XIAO Xue-mei LIU College of Science Civil Aviation University of China Tianjin 300300 China

Let ASG（2v ＋ l, v; Fq） be the （2v ＋l）-dimensional affine-singular symplectic space over the finite field Fq and ASp2v＋l,v（Fq） be the affine-singular symplectic group of degree 2v ＋ l over Fq. Let O be any orbit of flats under ASp2v＋l,v（Fq）. Denote by J the set of all flats which are joins of flats in O such that O LJ and assume the join of the empty set of flats in ASG（2v ＋ l, v;Fq） is φ. Ordering LJ by ordinary or reverse inclusion, then two lattices axe obtained. This paper firstly studies the inclusion relations between different lattices, then determines a characterization of flats contained in a given lattice LJ, when the lattices form geometric lattice, lastly gives the characteristic polynomial of LJ.

关键词： Lattice affine-singular symplectic groups characteristic polynomial

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利用计算机计算n阶图的特征多项式的方法

高等学校计算数学学报 2012年第3期34卷 260-266页

作者：刘木伙柳柏濂华南农业大学数学系华南师范大学数学科学学院南京师范大学数学科学学院

1引言设G=（V,E）为n阶无向的简单连通图.记N（v）为v的所有相邻点的集合,则d（v）=|N（v）|称为顶点v的度.若d（v）=1,则称v为G的一个悬挂点.设D（G）=diag(d（v;）,d（v;）,…,d（v;）)和A（G）分别表示图G的度对角矩阵和邻接矩阵,则L... 详细信息

1引言设G=（V,E）为n阶无向的简单连通图.记N（v）为v的所有相邻点的集合,则d（v）=|N（v）|称为顶点v的度.若d（v）=1,则称v为G的一个悬挂点.设D（G）=diag(d（v;）,d（v;）,…,d（v;）)和A（G）分别表示图G的度对角矩阵和邻接矩阵,则L（G）=D（G）-A（G）称为图G的Laplace矩阵,而Q（G）=D（G）+A（G）称为图G的SignlessLaplace矩阵.用符号N;表示一个m行n列的矩阵,M;表示一个n阶的方阵.特

关键词： characteristic polynomial Adjacency matrix Laplacian matrix Signless Laplacian matrix

The RCH Method for Computing Minimal polynomials of polynomial Matrices

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Journal of Systems Science & Complexity 2015年第1期28卷 190-209页

作者： YU Bo ZHANG Jintao XU Yanyan School of Mathematical Sciences Dalian University of Technology School of Science Dalian University of Technology

In this paper,a randomized Cayley-Hamilton theorem based method(abbreviated by RCH method) for computing the minimal polynomial of a polynomial matrix is *** determines the coefficient polynomials term by term from lower to higher *** using a random vector and randomly shifting,it requires no condition on the input matrix and works with probability *** the case that coefficients of entries of the given polynomial matrix are all integers and that the algorithm is performed in exact computation,by using the modular technique,a parallelized version of the RCH method is also *** with other algorithms in both theoretical complexity analysis and computational tests are given to show its effectiveness.

关键词： characteristic polynomial minimal polynomial polynomial matrix.

Quadratic Starlike Trees

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Algebra Colloquium 2023年第4期30卷 615-638页

作者： Yarong Hu Qiongxiang Huang College of Mathematics and System Science Xinjiang UniversityUrumqi 830046China School of Mathematics and Information Technology Yuncheng UniversityYunchengShanxi 044000China

We introduce the notion of a quadratic graph,which is a graph whose eigenvalues are integral or quadratic algebraic integral,and we determine nine infinite families of quadratic starlike trees,which are just all the quadratic starlike trees including integral starlike ***,the quadratic starlike trees are completely *** expressions for characteristic polynomials of quadratic starlike trees are also given.

关键词： quadratic algebraic integer starlike tree characteristic polynomial

Analysis of the effect on control systems of order variation for fractional-orderPI~λD~μcontrollers

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Journal of Harbin Institute of Technology(New Series) 2005年第3期12卷 336-341页

作者：曾庆山曹广益朱新坚 School of Electric Engineering Zhengzhou University Zhengzhou 450002China Dept. of Automation Shanghai Jiaotong University Shanghai 200030 China

This paper is concerned with fractional-order PI~λD~μcontrollers. The definitions and properties of fractional calculus are introduced. The mathematical descriptions of a fractional-order controller and fractional-order control systems are outlined. The effects on control systems of order variation for fractional-order PI~λD~μ controllers are investigated by qualitative analysis and simulation. The conclusions and simulation examples are given. The results show the fractional-order PI~λD~μ controller is not sensitive to variation of its order.

关键词： fractional calculus fractional-order control systems fractional-order PI~λD~μ controller characteristic polynomial

On Extending Real Functions to Finite Dimensional Algebras

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Acta Mathematica Sinica,English Series 1999年第4期15卷 455-470页

作者： Bingren Li Institute of Mathematics,Academia Sinica,Beijing 100080,P.R.China E-mail:libr@sun.ihep,ac.cnYimin Yang Division of Science & Mathematics,University of Minnesota,Mcrros,MN 56276 U S A E-mail:yangy@mrs,umn.edu Institute of Mathematics Academia Sinica Beijing P. R. China Division of Science & Mathematics University of Minnesota Mcrros USA

In a finite dimensional real algebra,we define functional calculus for real *** analytic hypothesis is replaced by a differentiable *** order differentibility is essentially necessary and sufficient to our ***,we give... 详细信息

关键词： m-Extension Continuous derivative characteristic polynomial

Factorizations of Matrices over Projective-free Rings

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Algebra Colloquium 2016年第1期23卷 23-32页

作者： I-Iuanyin Chen H. Kose Department of Mathematics Hangzhou Normal University Hangzhou 310036 China Department of Mathematics Ahi Evran University Kirsehir Turkey

An element of a ring R is called strongly J#-clean provided that it can be written as the sum of an idempotent and an element in J#（R） that commute. In this paper, we characterize the strong J#-cleanness of matrices over projective-free rings. This extends many known results on strongly clean matrices over commutative local rings.

关键词： strongly J#-matrix characteristic polynomial projective-free ring