求矩阵的秩、矩阵的秩在初等变换下的不变性pdf.pdf

文档介绍：

M2442


n
S

S
(
M2432,2433)

A
A
A
A



1 Æ A
F
n
A
⇔ A
0. ⇔ A

|A| =0 ⇔ A
n. 2

1 Æ A =(a )
F
m × n
r ≤ m
r ≤ n.

ij m×n

A
r
r
r
A ,
A
A

1 1
r
. 2
1 ≤ i <...< i ≤ m,1≤ j < ···<j ≤ n,
A
i ,...,i
Æ 1 r 1 r
 1 r
i1 i2 ··· ir
j ,...,j
A :

1 r
j1 j2 ··· jr
 
 a a ··· a 
 i1j1 i1j2 i1jr 

  
i i ··· i  a a ··· a 
1 2 r  i2j1 i2j2 irjr 
A =  
 . . . . 
j1 j2 ··· jr  . . . . 
 
 
airj1 airj2 ··· airjr
2 Æ A =(a )
F
A
r

ij m×n
0,
A
≥ r.

Æ 1 ≤ i < ··· <i ≤ m, 1 ≤ j < ··· <j ≤ n, A
i ,...,i

1 r 1 r 1 r

r
A
|A | =0.
j1,...,jr
1 1
A
A
A
j ,...,j

1 1 1 r
A
≥ r.

A
A
i ,...,i
A1
1 1 r
A
≥ r. 2
3 Æ
F
A =(a )
r,
A

ij m×n

r. 2
1
A
r.

A
r
A
r +1
r +1
0.
A
i ,...,i
j ,...,j
r