2 广义逆矩阵?A
1定义
., AGAG 记为的广义逆矩阵为则称
))(( ARbbAG b
,,mnnm CGCA 如果存在矩阵设使得
1定理使其满足充要条件是存在,mnCG
存在广义逆矩阵的则设 ACA nm,
AA G A?
pro of 必要性,nCu )( ARAub
AGb b? A G bA G A u? b? Au?
的一个广义逆矩阵为 AG
AA G A?
充分性,bAx? AGb AG Ax?
Ax? b?
pro of )()( AAAr a n kAr a n k )( AAra n k )( Aran k
},|{}1{ mnCGAAGAGA定义则有
2定理 的任一广义逆矩阵，是且设 AACA nm,
},|{}1{ mnCUAUAAAUAGGA
1推论 的一个广义是且设 ACACA mnnm,
)()( Ar a n kAr a n k
则逆矩阵,
{ | ( ) ( ),,}nmnmG G A E A A V W E A A V W C
proof }1{AG
AAAGAAAGAG )(U G A
{ 1 } { |,}nmA G G A U A A U A A U C
AA G A?
G A U A A U A A
mnCU
A U A AAU A AU A AUAG
G A U A A U A A
( ) ( )nmG A E A A U A A U E A A
)()( AAEWVAAEAG mn,WU?V U A A
},
),()(|{
mn
mn
CWV
AAEWVAAEAGG



},|{ mnCUUAAAUAGG
},
),()(|{
mn
mn
CWV
AAEWVAAEAGGM



)()( AAEWVAAEAM mn,nmV W C
AAAEWVAAEAAA M A mn )]()([
)()( AAAAAWVAAAAAAAAAM A
AAAAWVAAAAAM A )()(
),()(|{ AAEWVAAEAGG mn
},mnCWV }1{A
HHTT AAAAi )()(,)()()(
3定理 则设,,CCA nm
且都是幂等矩阵与,)( AAAAii
)()()( AAr a n kAAr a n kAr a n k
proof
其中的广义逆矩阵为,)( AAiii


0
00
1
的广义逆矩阵；是则且阶可逆矩阵是阶可逆矩阵是设
BSATSA TB
nTmSiv
11,
,,)(

);()(),()()( ANAANARAARv
AAAAi)( TTTT AAAA )(
)()( TT AA )()( HH AA同理可证
AAAAAAii 2)()(
)()( AAr a n kAr a n k
是幂等矩阵?AA
))()(( AAA0)(iii
的广义逆矩阵为 AA
AAAA )( AA
)( AAAra n k
)( Aran k? )()( AAr a n kAr a n k
0? A
0 ))()(( AAA AAA )(
A
BSABTiv 11)( ATSA A
的广义逆矩阵；是 BSAT 11BS A T
S ATSAS AT T 11
)()(),()( ANAANARAAR
)()(),()()( ANAANARAARv显然有
)()()( AAr a n kAAr a n kAr a n k又;)()( nEAAnAr a n ki的充要条件是
2推论 则设,nmCA
.)()( mEAAmAr a n kii的充要条件是充分性,)(3 ii定理 )()( AAr a n kAr a n k
nEra n k n )(
:证
nAr a n kAAr a n k )()(
阶可逆矩阵是 nAA? 1))(( AAAAE n
1))(( AAAAAA 1))(( AAAAAA
:证 P AQP AQP AQP AQ)(
必要性：
AA
}1{)( APPAQQ
1引理 都是设 nnmmnm CQCPCA,,
则可逆矩阵,
AAPPA QQA ])([ }1{)( APPAQQ
2引理 满足，则存在设 2112
22
11,XX
A
A
A


使得,0,0 112122221211 AXAAXA
}1{
2221
1211 A
AX
XA?
:证




22
11
2221
1211
22
11
2221
1211
0
0
0
0
A
A
AX
XA
A
A
A
AX
XAA



22
11
22222122
12111111
0
0
A
A
AAXA
XAAA

222222112122
221211111111
AAAAXA
AXAAAA
A
A
A


22
11
0
0
，则设


2221
1211
AA
AA
A4定理满足则存在存在如果 2112111,,)( XXAi?
，0)(,0)( 2112111212212111212212 XAAAAAAAAX
使得




rm
r
rn
r
EAA
E
AAAAX
XA
E
AE
1
112112111212221
12
1
11
1
11 0
)(0
}1{A
满足则存在存在如果 2112122,,)( YYAii?
，0)(,0)( 1221122121121122121121 YAAAAAAAAY
使得




rm
r
rn
r
E
AAE
AY
YAAAA
EAA
E
0
)(0 12111
1
2221
1221
1
221211
21
1
22
}1{A


rn
r
rm
r
E
AE
AA
AA
EAA
E
0
0 111
2221
1211
1
1121

12
1
112122
11
0
0
AAAA
A
proof