矩阵部分 一些知识回顾

1. 吉林大学国家精品课程 高等代数http://blog.sina.com.cn/s/blog_49ce771a01008dnq.html~type=v5_one&label=rela_nextarticle

3.9节有关于矩阵满秩分解的介绍: A=G*H

2.sylvester公式的证明

A is s*n matrix and B is n*m matrix, then rank(AB)>=r(A)+r(B)-n

Proof:
matirx C=
A 0
E B
has rank r(C)>=r(A)+r(B) (it can be proved by treating matrix as row vectors and using the definition of linear independence)

note that C has the same rank with D=
0 -AB
E 0
in fact, for full-rank matrix P and Q
P=[E, -A; 0,E] and Q=[E -B; 0 E], D=P*C*Q.

上面的网络教案中有利用满秩分解来证明的。