矩阵core - EP逆和DMP逆的广义Cayley - Hamilton定理
In this paper, we use the following notations.The symbol Cm,n is the set of m×n matrices with complex entries, and rk(A)represents the rank of A∈Cm,n.Let A∈Cn,n, and then the smallest non-negative integer k, which satisfies rk(Ak+1)=rk(Ak), is called the index of A and is denoted as Ind(A).The Moore-Penrose inverse of A∈Cm,n is defined as the unique matrix X∈Cn,m satisfying the equations AXA=A, XAX=X, (AX)*=AX, (XA)*=XA, and is denoted as X=A+.The Drazin inverse of A∈Cn,n is defined as the unique matrix X∈Cn,n satisfying the equations XAk+1=Ak, XAX=X, AX=XA and is usually denoted as X=AD (see Ref.[1]).The core-EP inverse of A∈Cn,n is defined as the unique matrix X∈Cn,n satisfying the equations XAk+1=Ak, XAX=X and (AX)*=AX, and is denoted as X=A⨁[2].The DMP inverse of A∈Cn,n is defined as the unique matrix X∈Cn,n satisfying the equations XAX=X, XA=ADA and AmX=AkA+, and is denoted as X=Ad,+[3].More details of the Drazin, core-EP, DMP inverses can be seen in Refs.[4-8].
The Cayley-Hamilton theorem has many applications in nonlinear time-varying systems, electric circuits, etc.The classical Cayley-Hamilton theorem was extended to the fractional continuous-time and discrete-time linear systems[9], nonlinear time-varying systems with square and rectangular systems[10], the Drazin inverse matrix and standard inverse matrix[11], etc.More details about the Cayley-Hamilton theorem and its applications can be read in Refs.[9-13].Therefore, it is very interesting to investigate the Cayley-Hamilton theorem for the core-EP inverse matrix and DMP inverse matrix.In this paper, our main tools are core-EP decomposition and generalized inverses.
1 Preliminaries
In this section, we present some preliminary results.
Theorem 1[14, Cayley-Hamilton theorem] Let pA(s)=det(sEn-A)be the characteristic polynomial of X∈Cn,n.Then pA(A)=0.
Theorem 2[14] Let A∈Cn,n is singular, i.e.det(A)=0, and the characteristic polynomial of A be
pA(s)=det(sEn-A)=sn+an-1sn-1+…+a1s
(1)
Then
fA(AD)=a1(AD)n+a2(AD)n-1+…+an-1(AD)2+AD=0
(2)
Lemma 1[15, core-nilpotent decomposition] Let A∈Cn,n be with Ind(A)=k.Then A can be written as the sum of matrices and i.e. where is nilpotent, and Here one or both of and can be null.Furthermore, there is a nonsingular matrix P such that
基于传统建筑工程施工中,各项工作均独立完善,极易导致信息不对称现象的发生,如若这一现象发生便会工程施工造成不利影响,在这种情况下通过BIM技术的运用可使数据不对称问题得到解决,可提高数据的精确度,从而不难发现,BIM技术可提高数据计算的灵活性及有效性,在土建工程施工中运用BIM技术可减少施工成本,确保工程工作效率的提升。将BIM技术运用到建筑工程施工中可起到三维渲染、快速算量的作用,详情如下:
(3)
where Σ∈Crk(A),rk(A) is non-singular, and is nilpotent and
Lemma 2[15, core-EP decomposition] Let A∈Cn,n be with Ind(A)=k.Then A can be written as the sum of matrices A1 and A2, i.e.A=A1+A2 where Ind(A1)≤1, A2 is nilpotent, and A2=A2A1=0.Here one or both of A1 and A2 can be null.Furthermore, there is a unitary matrix U such that
高校对“一带一路”沿线国家来华留学生进行教育必须依托于专业的课程设置。但较为遗憾的是,部分高校在“一带一路”沿线国家来华留学生的课程设置方面并不专业,甚至很多高校并未将“一带一路”沿线国家来华留学生课程与其它国家和地区的留学生课程有机区分开来。这样的做法并不能凸显高校在“一带一路”沿线国家来华留学生教育方面的针对性和区别性。不利于“一带一路”沿线国家来华留学生教育质量的提升。访谈中一名“一带一路”沿线国家来华留学生如此说道:“平常我们都是与其它国家的留学生一起上课的,并未听说学校对来自‘一带一路’国家的留学生在课程设置方面有所区别。”
(4)
where T∈Crk(A),rk(A) is non-singular, N is nilpotent and Nk=0.
Let the core-EP decomposition of A be as in (4).Then
A⨁
(5)
Lemma 3 Let the core-EP decomposition of A∈Cn,n be as in Lemma 2.Then the core-EP inverse of A is
(6)
and
(A⨁
(7)
where It is easy to confirm that Φj=Tj-kΦk, where j≥k.
2 Main Results
红土镍矿主要分为褐铁矿型和硅镁镍矿型两种。褐铁矿类型红土镍矿组成特点是:含Fe较高,一般40%~50%,MgO 0~5% , SiO2 10%~30%;硅镁镍矿型红土镍矿组成特点是:含Fe较低,一般15%~30%,含MgO 15%~35%,SiO2 10%~30%。采用还原熔炼工艺后,由于该法属于熔池熔炼,可通过改变炉内的还原氛围实现镍铁的选择性还原性。由于金属镍熔点为1 450 ℃,冶炼熔渣温度必须在该温度以上。
(1)教师可以更加关注学生主体。任何课堂都应当是以学生为主体,教学内容和活动的创设都是服务于学生的认知。教师在教学准备、教学过程中时刻关注学生,不预设学生的回答。在教学中,对学生的知识储备不足或前概念的偏差而带来的问题,教师应当及时发现并纠正。比如,在“植物的生殖”一课中,学生观看完嫁接标准示范视频后,一名学生指出了其中一名学生代表操作示范的一个错误点,教师就应当深入挖掘,请其他学生指出更多操作示范错误点,以此达到巩固嫁接和扦插这两种技术手段操作要领的作用。
Lemma 4 Let the characteristic polynomial of A∈Cn,nbe as in Eq.(1).Then
上述方法的缺点导致用户花费时间和精力却筛选不到更符合自身需求的服务。对语义和数值进行综合考量并考虑服务请求者的个性化需求,提出了一种基于语义和数值综合匹配的Web服务选择方法,包含QoS语义和数值综合匹配、构建多属性决策矩阵、个性化服务选择这几个阶段。
fA(A⨁)=a1(A⨁)n+a2(A⨁)n-1+…+
an-1(A⨁)2+A⨁=0
(8)
Proof Using (1)and Theorem 1 we obtain
An+an-1An-1+…+a1A=0
(9)
It follows from Lemma 2 and (6)that
(10)
Post-multiplying (10)by
we have
(11)
Therefore, by applying (7), we obtain (8).
Example 1 Let
In this section the classical Cayley-Hamilton theorem will be extended to the core-EP inverse matrix and DMP inverse matrix.By assumption, matrix A is singular, i.e.det(A)=0.
解析逻辑关系:从施工过程上看,依次为:空间形象,室内装修,室内物理环境,室内陈设艺术,各个班组连续作业,依次进行,且没有间断。在相邻过程上,每个工程分别投入的时间是:8天,12天,4天,8天。每个工程分别结束的时间是第8天,第20天,第24天,第28天.从此施工进展来看,只要保证第一个施工过程正常投入,即可满足随后的过程连续施工和依次搭建,即该施工过程逻辑关系准确。
and the core-EP inverse A⊕ is
A⊕
Proof Let the core-EP decomposition of A, i.e.A=A1+A2, be as in Lemma 2.Then
s4-2s3+s2+0s
From the classical Cayley-Hamilton theorem, we have A4-2A3+A2=0.By applying Lemma 4, we obtain (A⊕)3-2(A⊕)2+A⊕=0.
Note that, if the characteristic polynomials of A⨁ and AD is
在以上工程设计与施工环节的工程造价管理中合理应用BIM技术,可进一步丰富BIM模式中所涵盖的信息量,而这一点也可极大地控制工程验收过程中可能会出现的纠纷。除此之外,BIM技术模型也可以在验收阶段提供完整的结算资料,便于对工程建设各个环节的相关数据进行审查。
pA⨁(s)=det(sEn-A⨁)=sn+bn-1sn-1+…+b1s
(12)
pAD(s)=det(sEn-AD)=sn+cn-1sn-1+…+c1s
(13)
respectively, we cannot obtain
pA⨁(s)=b1An+b2An-1+…+bn-1A2+A
(14)
pAD(s)=c1An+c2An-1+…+cn-1A2+A
(15)
It is easy to confirm that the core-EP inverse A⊕ is
Example 2 Let
A⊕
Then
but
fAD(s)=fA⊕(s)=0A2+A=A≠0
Theorem 3 Let A∈Cn,n and Ind(A)=k.Then the characteristic polynomial of A⨁∈Cn,n is
Then Ind (A)=2, the Drazin AD is
pA⨁(s)=sn+bn-1sn-1+…+bn-rk(Ak)sn-rk(Ak)
(16)
Furthermore,
bn-rk(Ak)An+…+bn-1An-rk(Ak)+1+An-rk(Ak)=0
(17)
The characteristic polynomial of A is
sn-rk(Ak)det(sErk(Ak)-T-1)
Therefore, we obtain (16).Using (16)and Theorem 1, we obtain
(A⨁)n+bn-1(A⨁)n-1+…+bn-rk(Ak)(A⨁)n-rk(Ak)=0
(18)
that is,
(19)
Post-multiplying (19)by
we have (17).
Theorem 4 Let A∈Cn,n, Ind(A)≤1 and the characteristic polynomial of A⨁∈Cn,n be as in (12).Then
fA⨁(A)=b1An+b2An-1+…+bn-1A2+A=0
(20)
Theorem 5 Let A∈Cn,n and Ind(A)=k.Then the characteristic polynomial of AD∈Cn,n is
pAD(s)=det(sEn-AD)=sn+cn-1sn-1+…+cn-rk(Ak)sn-rk(Ak)
(21)
Furthermore,
cn-rk(Ak)An+…+cn-1An-rk(Ak)+1+An-rk(Ak)=0
(22)
Proof Let the core-nilpotent decomposition of A, be as in Lemma 1.Then
sn-rk(Ak)det(sErk(Ak)-Σ-1)
Therefore, we obtain (21).Using (21)and Theorem 1 we obtain
(AD)n+bn-1(AD)n-1+…+bn-rk(Ak)(AD)n-rk(Ak)=0
that is,
Post-multiplying the above equation by
(2)教学文件齐备,但大纲的执行情况较差,难以保证教学内容的系统性,教学管理部门应加强监督力度,进一步规范管理。
Therefore, we obtain (22).
Theorem 6 Let A∈Cn,n and Ind(A)≤1 and the characteristic polynomial of AD∈Cn,n be as in (13).Then
fAD(A)=c1An+c2An-1+…+cn-1A2+A=0
(23)
Let A∈Cn,n and Ind(A)=k.Then the DMP inverse of A is Ad,+=ADAA+[3].Since (Ad,+)2=ADAA+ADAA+=ADAA+AADA+=(AD)2AA+, we obtain
(Ad,+)p=(AD)pAA+
(24)
where p is a positive integer.
Theorem 7 Let the characteristic polynomial of A∈Cn,n be as in (1).Then
fA(Ad,+)=a1(Ad,+)n+a2(Ad,+)n-1+…+
an-1(Ad,+)2+Ad,+=0
(25)
Proof By applying (1)and Theorem 2, we obtain
a1(AD)nAA++a2(AD)n-1AA++…+
an-1(AD)2AA++ADAA+=0
From (24), we can obtain (25).
“没个正形。”思蓉说,“我和你姐夫还真以为你们出了什么事情,下节目就一起跑过来,饭都没顾上吃。家里有吃的没有?”
References
[1]Ben-Israel A, Greville T N E.Generalized inverses: Theory and applications[M].2nd ed.Berlin: Springer, 2003: 163-168.
[2]Manjunatha Prasad K, Mohana K S.Core-EP inverse[J].Linear and Multilinear Algebra, 2014, 62(6): 792-802.DOI:10.1080/03081087.2013.791690.
[3]Malik S B, Thome N.On a new generalized inverse for matrices of an arbitrary index[J].Applied Mathematics and Computation, 2014, 226: 575-580.DOI:10.1016/j.amc.2013.10.060.
[4]Gao Y, Chen J, Ke Y.*-DMP elements in *-semigroups and *-rings[J].arXiv preprint.arXiv: 1701.00621, 2017.
[5]Li T, Chen J.Characterizations of core and dual core inverses in rings with involution[J].Linear and Multilinear Algebra, 2017: 1-14.DOI: 10.1080/03081087.2017.1320963.
[6]Wang H.Core-EP decomposition and its applications[J].Linear Algebra and Its Applications, 2016, 508: 289-300.DOI:10.1016/j.laa.2016.08.008.
[7]Zou H, Chen J, Patrìcio P.Characterizations of m-EP elements in rings[J].Linear and Multilinear Algebra, 2017: 1-13.DOI: 10.1080/03081087.2017.1347136.
[8]Prasad K M, Mohana K S.Core-EP inverse[J].Linear and Multilinear Algebra, 2014, 62(6): 792-802.DOI:10.1080/03081087.2013.791690.
[9]Kaczorek T.Cayley-Hamilton theorem for fractional linear systems[C]//8th Conference on Non-Integer Order Calculus and Its Applications.Zakopane, Poland, 2017: 45-56.DOI:10.1007/978-3-319-45474-0_5.
[10]Kaczorek T.An extension of the Cayley-Hamilton theorem for nonlinear time-varying systems[J].International Journal of Applied Mathematics and Computer Science, 2006, 16: 141-145.
[11]Kaczorek T.Cayley-Hamilton theorem for Drazin inverse matrix and standard inverse matrices[J].Bulletin of the Polish Academy of Sciences Technical Sciences, 2016, 64(4): 793-797.DOI:10.1515/bpasts-2016-0088.
[12]Feng L G, Tan H J, Zhao K M.A generalized Cayley-Hamilton theorem[J].Linear Algebra and Its Applications, 2012, 436(7): 2440-2445.DOI:10.1016/j.laa.2011.12.015.
[13]Hwang S G.A generalized Cayley-Hamilton theorem for polynomials with matrix coefficients[J].Linear Algebra and Its Applications, 2011, 434(2): 475-479.DOI:10.1016/j.laa.2010.08.039.
[14]Horn R A, Johnson C R.Matrix analysis[M].Cambridge,UK: Cambridge University Press, 2012: 109-111.
[15]Wang H, Liu X.Partial orders based on core-nilpotent decomposition[J].Linear Algebra and its Applications, 2016, 488: 235-248.DOI:10.1016/j.laa.2015.09.046.
下一篇:没有了