必要性：R( A) R( AH )uAuAuuuuuPuuRu(uAu)u,uAuuuAuuuuPuRu(uAuHuur) AA A A
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定理 4 设 A C mn，则有 (1) ( AH A) A ( AH ) ,( AAH ) ( AH ) A; (2) ( AH A) A ( AAH ) A AH ( AAH ) ( AH ); (3) AA ( AAH )( AAH ) ( AAH ) ( AAH );
返回
证:(1) r 0 A 0 A 0 (2) r 0 存在最大秩分解A BD,(B Crmr , D Crrn ) BH B Crrr , DDH Crrr G D1H4(2DD4H3)11(B4H2B)413BH
D的公式法构造的右逆 B的公式法构造的左逆
AGA A GAG G ( AG)H G H AH [DH (DDH )1(BH B)1 BH ]H (BD)H
A[( AAH ) ]H A A ( AAH ) A (3) AA A [ AH ( AAH ) ] ( AAH )( AAH ) (I )
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( AAH ) ( AAH ) (BDDH BH ) (BDDH BH ) BDDH (DDH BH BDDH )1(BH B)1 BH BDDH BH BDDH (DDH )1(BH B)1(DDH )1(BH B)1BH BDDH BH BDDH (DDH )1(BH B)1(DDH )1 DDH BH BD[DH (DDH )1(BH B)1 BH ]
A ABBH AH ABB
,
为Hermite矩阵
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B[(BH B)1]H [(DDH )1]H DDH BH
返回
( AG)H B[(BH B)1]H [(DDH )1]H DDH BH B[(BH B)H ]1[(DDH )H ]1 DDH BH B(BH B)1(DDH )1 DDH BH
B(BH B)1 BH BDDH (DDH )1(BH B)1 BH AG (GA)H AHG H DH BH B[(BH B)1]H [(DDH )1]H D DH BH B(BH B)1(DDH )1 D DH (DDH )1 D DH (DDH )1(BH B)1 BH BD GA G是A的M P广义逆矩阵A
A A ( AH A)( AH A) ( AH A) ( AH A).
证: (1) A BD是最大秩分解
A DH (DDH )1(BH B)1 BH
( AH A) (DH BH BD) { DH 1B44H2B4D43为AH A最大秩分解 uuB1uuuuuDu1uuuuuuuuuuuuuuuuuuuuuuru
( AH A) (BH BD)H (BH BDDH BH B)1(DDH )1 D 返回
DH BH B(BH B)1(DDH )1(BH B)1(DDH )1 D DH (DDH )1(BH B)1 BH B(BH B)1(DDH )1 D [DH (DDH )1(BH B)1 BH ][B(BH B)1(DDH )1 D] A ( AH ) (2) ( AH A) A ( AH ) A ( A )H A[ AH ( AAH ) ]H
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证:(2) A BD为最大秩分解 BH B, DDH可逆 ( A )T [DH (DDH )1(BH B)1 BH ]T
(BH )T [BT (BH )T ]1[(DH )T DT ]1(DH )T (BT )H [BT (BT )H ]1[(DT )H DT ]1(DT )H ( AT )
AA (II )
(I ),(II )
AA ( AAH ) ( AAH )
( AAH )( AAH )
返回
定理 5 设A C ml，B B) R(B),
R(
BB
H
AH
)
R( AH
).
A ABBH AH
B
BAH
AB
BBH AH AH AB.
,
5 M - P广义逆矩阵A+
定义 1 设 A C mn，如果有G C nm，使得 AGA A, GAG G, (GA)H GA,( AG)H AG,
则称G是A的M P广义逆矩阵，记为G A .
定理 1 设A Crmn, A BD是A的最大秩分解,则
G DH (DDH )1(BH B)1 BH 就是A的M P广义逆矩阵A .
最大秩分解
返回
uAuHuuAuuuuBu1uDuu1u;uuBuu1uuuDuuHuu,uDu1uuuuBuHuuBuuDuru
( AH A) AH [D1H (D1D1H )1(B1H B1)1B1H ](BD)H (BH BD)H [(BH BD)(BH BD)H ]1(DDH )1 D(BD)H DH BH B(BH BDDH BH B)1(DDH )1 DDH BH DH BH B(BH B)1(DDH )1(BH B)1(DDH )1 DDH BH
uRu(uAuuAuuu) uuuRu(uAuu)u, uRu(uAuuuAuu)uuuRu(uuAuu)u,uRuu(uAuuu) uuuRu(uAuuHuur)u AA =PR( A), A A PR( AH ) (6)充分性：AA =A A，A是A的自反广义逆矩阵
R( A) R( AA ) =R( A A) R( A ) R( AH )
(3) A BD为最大秩分解 AH A (BD)H BD
DH BH BD DH (BH BD) uBu1uuuDuuHuu,uDuu1uuuBuuHuBuuDur
AH A B1D1 rank( A) rank( AH A) r r rank(D) rank(B1)
rank(BH BD) rank(D1) AH A B1D1是AH A的
DH (DDH )1(BH B)1 BH A (4) A是A的自反广义逆 rank( A ) rank( A) rank( AH )uRu(uAuuu)uuuuRu(uAuuHuur) R( A ) R( AH )
返回
(5) A是A的自反广义逆 AA和A A是幂等矩阵 AA =PR( AA ) , A A PR( A A)
返回
定理 2 设 A C mn，则A是唯一的.
定理 3 设 A C mn，则有 (1) ( A ) A; (2) ( AT ) ( A )T ,( AH ) ( A )H ; (3) A ( AH A) AH AH ( AAH ); (4) R( A ) R( AH ); (5) AA PR( A), A A PR( AH ); (6) R( A) R( AH ) AA A A.