(2)λ=0时,A=0nn,A'A=0m,显然成立 λ≠0时,(A)A)(A)=(44)(AAA)=2A ()(SAT(TAS-)(SAT)=S(AAA)T=SAT (4)rank(A)=rank(AAOAsrank(AO) AAOAAO=A.AC (5)AA(A=A AA)=A 和,AA=A.4→( 又 rank(A)=ramk(A4")A)≤runk(AA)≤ramk(4) →rnk(A40)=ramk(A) 同理,rmk(AA)=rmk(A (6)·R(A)={Ax|x∈C"}cCm,R(A4)={A4y|y∈Cm}c R(A)2R(AAO)R(AA A)>R(A)=R(AA) 同理 又法:将A,AA写成A=[aa2…an],AA=[bb2…bn a1,b,均为m维列向量,则 R(4)=spm{a,a2…an}=2∑5|5∈C}→dimR(A)=rmk(A R(A4)=mb…bn}=①n|∈C}→dimR(A1)=rmk(A4") 即dimR(A)= dim r(AA)且R(A)2R(AA) 故R(4)=R(A4) N(A)={x|Ax=0,x∈C"}≤C",N(型A={x|AAx=0,x∈C"}≤CN →N(A)gN(AA)N(AAA)=N(A)→N(A)=N(AA) 同理 又法:dimN(A)=n-rmk(A)=n-rmhk(AA=dimN(AA)(2)  = 0 时， 0 A＝ m n ， † (1) 0  A = n m .显然成立.   0 时， † (1) 1 (1) ( )( )( ) ( )( )       A A A AA A A − = = (3) 1 (1) 1 (1) ( )( )( ) ( ) SAT T A S SAT S AA A T SAT − − = = (4) (1) (1) rank A rank AA A rank A ( ) ( ) ( ) =  (5) (1) (1) (1) (1) 2 (1) (1) (1) (1) (1) (1) 2 (1) ( ) ( ) AA A A A A AA AA AA A A A AA A A A A A A A   =  → = = →    =  → = 又 (1) (1) (1) ( ) ( ) ( ) ( ) ( ) ( ) rank A rank AA A rank AA rank A rank AA rank A =   → = 同理， (1) rank A A rank A ( ) ( ) = (6) • ( ) { | }n m R A Ax x C C =   , (1) (1) ( ) { | }m n R AA AA y y C C =   (1) (1) (1) R A R AA R AA A R A R AA ( ) ( ) ( ) ( ) ( )     → = 同理 又法：将 (1) A,AA 写成 (1) 1 2 1 2 [ ], [ ]. A a a a AA b b b = = n m , i j a b 均为 m 维列向量，则 1 2 1 ( ) { , , , } { | } dim ( ) ( ) n n i i i i R A span a a a a C R A rank A   = = =  → =  (1) (1) (1) 1 2 1 ( ) { , , , } { | } dim ( ) ( ) m n i i i i R AA span b b b b C R AA rank AA   = = =  → =  即 (1) (1) dim ( ) dim ( ) ( ) ( ) R A R AA R A R AA =  且 故 (1) R A R AA ( ) ( ) = • ( ) { | 0, } , n n N A x Ax x C C = =   (1) (1) ( ) { | 0 , }n N N A A x A Ax x C C = =   (1) (1) (1) ( ) ( ) ( ) ( ) ( ) ( ) N A N A A N AA A N A N A N A A     = → = 同理 又法： (1) (1) dim ( ) - ( ) - ( ) dim ( ) N A n rank A n rank A A N A A = = =