上三角矩阵空间的保持逆矩阵的函数  

Function Preserving Inverses of Upper Triangular Matrices

作　　者：樊玉环[1] 袁海燕[1] 魏喆[1] FAN Yu-huan;YUAN Hai-yan;WEI Zhe(College of Science,Heilongjiang Institute of Technology,Harbin,Heilongjiang 150001,China)

机构地区：[1]黑龙江工程学院理学院,黑龙江哈尔滨150000

出　　处：《河北北方学院学报（自然科学版）》2021年第7期7-13,共7页Journal of Hebei North University:Natural Science Edition

基　　金：黑龙江省科技厅自然科学基金项目(LH2019A030);黑龙江工程学院创新团队项目(2018CX17)。

摘　　要：目的保持问题中的函数保持问题的基本思路主要有寻求新的不变量或者把已有结果的条件削弱或者改变已有结果所作用的集合,在全矩阵空间上保持逆矩阵的函数形式刻画的基础上改变所作用的集合为上三角矩阵空间,研究上三角矩阵空间上保持逆矩阵的函数的形式。方法以上三角矩阵空间中的逆矩阵为研究对象,通过线性代数中矩阵的运算及群论中同态的研究方法,寻找特殊的满足互逆的上三角矩阵,使定义在域上的上三角矩阵空间中2个互逆的矩阵经函数后,所得2个新的矩阵仍为互逆矩阵,从而建立上三角矩阵空间上保持逆矩阵的函数的形式。通过特殊矩阵的选取刻画出函数的形式。结果①f是n(n≥4)阶上三角矩阵空间的保持逆矩阵的函数的充要条件是f=δ,δ是域F上的满足δ(1)=1单的自同态。②f是T_(2)(F)保持逆矩阵的函数的充要条件是f是非零乘法奇函数。③f是T3(F)保持逆矩阵函数的充要条件是f=f^(-1)(1)δ,其中δ是域F上的满足δ=1单的自同态。结论上三角矩阵空间上保持逆矩阵的函数的形式已经给出,但是这里要求域的特征不为2,当域的特征为2时还需要进一步的研究。Objective The basic idea of function preserving may be to look for new invariants,or weaken the conditions of existing results,or change the set of existing results.On the basis of preserving the functional representation of inverse matrix on the whole matrix space,we change the matrix space into upper triangular matrix,and continue to study the function preserving inverses of upper triangular matrix.Methods The inverse matrices of the upper triangular matrix space were studied.By applying the knowledge of matrix operation in linear algebra and homomorphism in group theory,special reciprocal matrix of upper triangular matrices were found.After the function action,the two new matrices were still reciprocal matrix,and then the function form of preserving inverses of upper triangular matrices was established.Results①f is a function of preserving of upper triangular matrices(the order of matrix is greater than or equal to 4)if and only if f=δ,whereδis a single endomorphism in the field F andδsatisfiesδ(1)=1.②f is a function of preserving of upper triangular matrix of order 2 if and only if f is nonzero,whereδwas a multiplicative odd function.③f is a function of preserving inverse matrix of upper triangular matrix of order 3 if and only If f=f^(-1)(1)δ,whereδis a single endomorphism in the field F andδsatisfiesδ(1)=1.Conclusion The form of the function of preserving inverse matrix of upper triangular matrix is given.However,the characteristic of the domain can not be 2.When the feature of domain is 2,further research is needed.

关 键 词：上三角 保持 逆矩阵 函数 

分 类 号：O153.3[理学—数学]