PA=LU（带行交换的矩阵分解）
Do all the exchanges before elimination !!
矩阵A中主元位置出现0，我们使用置换矩阵对其进行行交换
P
21
A
=
[
0
1
0
1
0
0
0
0
1
]
[
0
1
1
1
2
1
2
7
9
]
=
[
1
2
1
0
1
1
2
7
9
]
P_{21}A= \begin{bmatrix} 0 & 1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 0 & 1 & 1 \\ 1 & 2 & 1\\ 2 & 7 & 9 \end{bmatrix}= \begin{bmatrix} 1 & 2 & 1\\ 0 & 1 & 1\\ 2 & 7 & 9 \end{bmatrix}
P21?A=???010?100?001???????012?127?119????=???102?217?119????
消元
a
31
a_{31}
a31?
M
u
l
t
i
p
l
i
e
r
?
l
31
=
a
31
a
11
=
2
Multiplier\ l_{31}=\frac{a_{31}}{a_{11}}=2
Multiplier?l31?=a11?a31??=2
n
e
w
?
r
o
w
3
=
r
o
w
3
?
l
31
r
o
w
1
new\ row3 = row3-l_{31}row1
new?row3=row3?l31?row1
E 31 P 21 A = [ 1 0 0 0 1 0 ? 2 0 1 ] [ 1 2 1 0 1 1 2 7 9 ] = [ 1 2 1 0 1 1 0 3 7 ] E_{31}P_{21}A= \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ -2 & 0 &1 \end{bmatrix} \begin{bmatrix} 1 & 2 & 1\\ 0 & 1 & 1\\ 2 & 7 & 9 \end{bmatrix}= \begin{bmatrix} 1 & 2 & 1\\ 0 & 1 & 1\\ 0 & 3 & 7 \end{bmatrix} E31?P21?A=???10?2?010?001???????102?217?119????=???100?213?117????
消元
a
32
a_{32}
a32?
M
u
l
t
i
p
l
i
e
r
?
l
32
=
a
32
a
22
=
3
Multiplier\ l_{32}=\frac{a_{32}}{a_{22}}=3
Multiplier?l32?=a22?a32??=3
n
e
w
?
r
o
w
3
=
r
o
w
3
?
l
32
r
o
w
2
new\ row3 = row3-l_{32}row2
new?row3=row3?l32?row2
E 32 E 31 P 21 A = [ 1 0 0 0 1 0 0 ? 3 1 ] [ 1 2 1 0 1 1 0 3 7 ] = [ 1 2 1 0 1 1 0 0 4 ] = U E_{32}E_{31}P_{21}A= \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & -3 &1 \end{bmatrix} \begin{bmatrix} 1 & 2 & 1\\ 0 & 1 & 1\\ 0 & 3 & 7 \end{bmatrix}= \begin{bmatrix} 1 & 2 & 1\\ 0 & 1 & 1\\ 0 & 0 & 4 \end{bmatrix}=U E32?E31?P21?A=???100?01?3?001???????100?213?117????=???100?210?114????=U
P A = E 31 ? 1 E 32 ? 1 U = L U PA=E_{31}^{-1}E_{32}^{-1}U=LU PA=E31?1?E32?1?U=LU
其中
E
31
?
1
E_{31}^{-1}
E31?1? 是
E
31
E_{31}
E31? 的逆过程，
E
32
?
1
E_{32}^{-1}
E32?1? 是
E
32
E_{32}
E32? 的逆过程
E
31
?
1
=
[
1
0
0
0
1
0
2
0
1
]
?
E
32
?
1
=
[
1
0
0
0
1
0
0
3
1
]
?
L
=
E
31
?
1
E
32
?
1
=
[
1
0
0
0
1
0
2
0
1
]
[
1
0
0
0
1
0
0
3
1
]
=
[
1
0
0
0
1
0
2
3
1
]
E_{31}^{-1}=\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 2 & 0 &1 \end{bmatrix}\\ ~\\ E_{32}^{-1}=\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 3 &1 \end{bmatrix}\\ ~\\ L=E_{31}^{-1}E_{32}^{-1}=\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 2 & 0 &1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 3 &1 \end{bmatrix}=\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 2 & 3 &1 \end{bmatrix}
E31?1?=???102?010?001?????E32?1?=???100?013?001?????L=E31?1?E32?1?=???102?010?001???????100?013?001????=???102?013?001????
