用广义逆矩阵方法对矩阵秩的一些讨论

DISCUSSION ON RANK OF MATRIX BY MEANS OF GENERALIZED INVERSE

本文给任意除环上的矩阵的秩提出了一个公理化定义,并且从定义出发证明了一些常见的矩阵秩的性质。整个讨论试图使用一些最初步的广义逆矩阵知识而避免使用向量的线性相关、线性无关及向量空间的基、维数等概念。能够这样做是因为通常讨论矩阵秩时使用的消元法或找齐次线性方程组解空间的方法,都可以用广义逆矩阵给出形式化的表示。

This paper gives an axiomatic definition to the rank of matrix over an arbitrary division ring and proves some common properties of rank of matrix on the basis of this definition. Some knowledge of generalized inverse is used. The definition is given as follows: A numerical function p(·) defined on a set of matrices is called rank function and 0(A), A∈ is called rank of matrix A if the following conditions are satisfied, (‘ρ1) ρ(A)=0, when A is a column vector and A=0; (ρ2) ρ(A)=1, when A is a column vector and A≠0; (ρ03) ρ(A, B)]=ρp(A)+ρ(I-AA~) B], where (A, B) are partitioned matrices and A- is any particular generalized inverse of A.