分块矩阵一、分块矩阵的概念
=
34333231
24232221
14131211
aaaa
aaaa
aaaa
A
=
2221
1211
AA
AA
=
34333231
24232221
14131211
aaaa
aaaa
aaaa
A
=
232221
131211
AAA
AAA
定义：将矩阵用若干纵横直线分成若干个小块，
每一小块称为矩阵的子块（或子阵），以子块为元素形成的矩阵称为分块矩阵。
二、分块矩阵的运算
1.线性运算加法与数乘
2.乘法运算符合乘法的要求
3.转置运算大块小块一起转
T
T
AAA
AAA
A
=
232221
131211
=
TT
TT
TT
AA
AA
AA
2313
2212
2111
三、几种特殊的分块阵
1.准对角阵
=
s
A
A
A
A
O
2
1
),,2,1( siA
i
L=为方阵，
准对角阵或分块对角阵
=
s
A
A
A
A
O
2
1
=
s
B
B
B
B
O
2
1
),,2,1,( siBA
ii
L=为同阶方阵，
BA±
±
±
±
=
ss
BA
BA
BA
O
22
11
kA
=
s
kA
kA
kA
O
2
1
AB
=
ss
BA
BA
BA
O
22
11
=
T
s
T
T
T
A
A
A
A
O
2
1
=
m
s
m
m
m
A
A
A
A
O
2
1
s
AAAAL
21
=
),,2,1( si
AA
i
L=
可逆可逆
=
1
1
2
1
1
1
s
A
A
A
A
O
)()()()(
21 s
ArArArAr +++=L
牢记这些公式！
例1
=
3100
3200
0010
0021
A
解：将矩阵分块求A的行列式，秩及逆。
=
2
1
A
A
A
21
AAA =?
3=
4)( =Ar
=
1
2
1
1
1
A
A
A
=
3
2
3
1
00
1100
0010
0021
只须口算即可！
2.分块三角阵
=
22
1211
AO
AA
A
分块上三角阵或准上三角阵
.)2,1( =iA
ii
为方阵，
2211
AAA =
.)2,1( =
i
AA
ii
可逆可逆
=

1
22
1
2212
1
11
1
11
1
AO
AAAA
A
=
2221
11
AA
OA
A
=

1
22
1
1121
1
22
1
11
1
AAAA
OA
A
=
2221
1211
1
XX
XX
A设则
=
22
1211
1
AO
AA
AA
2221
1211
XX
XX
=
EO
OE
++
=
22222122
2212121121121111
XAXA
XAXAXAXA
EXA =
2222
1
2222
=? AX
OXA =
2122
OX =?
21
EXAXA =+
21121111
1
1111
=? AX
OXAXA =+
22121211
1
2212
1
1112

=? AAAX
=
2000
1200
3120
4312
.2 A求矩阵的逆例解：将矩阵分块
=
22
1211
AO
AA
A
1
22
1
11
2
1
0
4
1
2
1

=
= AA
=

1
22
1
2212
1
11
1
11
1
AO
AAAA
A


=
2/1000
4/12/100
8/54/12/10
16/58/54/12/1
1
2212
1
11

AAA
只须计算
3.分块斜对角阵
=
OB
AO
M
可逆可逆BAM,?
=
OA
BO
M
1
1
1
=
0030
0021
2100
5300
.3 M求矩阵的逆例解：将矩阵分块
=
OB
AO
M
=
OA
BO
M
1
1
1
=
0031
0052
3/1000
3/2100
只须口算即可！