矩阵多项式秩的几个恒等式及其应用  

Several Rank Identities of Matrix Polynomials and Its Applications

作　　者：王廷明[1] 

机构地区：[1]青岛大学师范学院数学系,青岛266071

出　　处：《德州学院学报》2011年第6期1-4,共4页Journal of Dezhou University

基　　金：青岛大学教学研究项目(JY0937)

摘　　要：借助以矩阵多项式为系数矩阵的齐次线性方程组解空间的直和分解结果,给出了一般数域上矩阵多项式秩的几个基本恒等式.作为应用,得到了复数域上矩阵可对角化的一个充要条件,给出了复数域上线性空间关于其上的线性变换的准素分解定理的简洁证明.最后提出一个关于矩阵多项式秩等式的公开问题.Based on the results of the direct sum decomposition on the solution space of homogeneous linear equations which coefficient matrix is matrix polynomial, several identities of the rank of matrix polynomi als on general number field are presented. As applications, a sufficient and necessary condition is derived to diagonalizable of matrix on complex number field, and a concise proof is presented to primary decompo sition theorem of the linear space on complex number field with respect to the linear transformation on it. Finally, an open question of the rank identity of matrix polynomials is brought up.

关 键 词：矩阵秩 矩阵多项式 解空间 特征多项式 最小多项式 

分 类 号：O151.2[理学—数学]