关于矩阵多项式的交换性  

On the commutativity of matrix polynomials

作　　者：雷震 LEI Zhen(School of Mathematics and Statistics,Anhui Normal University,Wuhu 241000,China)

机构地区：[1]安徽师范大学数学与统计学院,安徽芜湖241000

出　　处：《高师理科学刊》2022年第7期29-31,共3页Journal of Science of Teachers＇College and University

基　　金：安徽省自然科学基金项目(2008085MA06);安徽省教育厅省级教学示范课项目(皖教秘高字[2020]165号)——高等代数(安徽师范大学);安徽师范大学数学与统计学院课程思政培育项目——线性代数。

摘　　要：设P是为数域,应用哈密尔顿-凯莱定理证明了:设B为n阶方阵,若存在n阶方阵A的多项式f(A),使得f(A)(B+bE)=E,则对于A的任意多项式g(A)及B的任意多项式h(B),有g(A)h(B)=h(B)g(A)成立,这里b为P中的元素,E为n阶单位矩阵.进一步地,当P为一个有单位元的结合的交换环时,结论仍然成立.根据线性方程组解的理论,证明了矩阵A的伴随矩阵A*的多项式及其逆矩阵都可以表示成A的多项式.Let P be a number field,by using Hamilton-Kelley theorem,it is proved that if B is an n-order square matrix,and there is a polynomial f(A) of A such that f(A)(B+b E) = E,then g(A)h(B) = h(B)g(A) for any polynomial g(A) of A and any polynomial h(B) of B,whereb∈P and E is an identity matrix.Furthermore,when P is an associative commutative ring with unity,the above conclusion is still true.In addition,according to the solution theory of linear equations,it is given that the polynomials and inverse matrices of the adjoint matrix Aof matrix A can be expressed as polynomials of A.

关 键 词：矩阵多项式 交换性 特征多项式 伴随矩阵 

分 类 号：O153.3[理学—数学]