b=∑9bm(=1,2,…,n) b AB b2_∑a2b 可见,AB的各行向量均为bn,bn…,b的线性组合,故 nk(AB)≤ rank B 合起来即rmnk(AB)≤min( GrankA, rank B) 定理:设A∈Cm,BEC",A∈C,=2≠0则 0A=0 (1)(A)∈A2{l} (2)AA∈(AA){} (3)S、T为可逆方阵且与A可乘,则 AS∈(SA7){l},(S∈Cmm,T∈C) (4)ramk(A)≥rmkA (5)AA和AA均为幂等矩阵且与垌秩(P2=P) (6)R(AAO)=R(A), N(AA)=N(A), R((AA)=R(A") (7)AA=I A4=1n台 rank(4)=m AB(AB)A=A rank(AB)=rank(A) B(AB)AB=B rank(AB)=rank(B) 证明:(1)(4)A2=(AAA)2 1}1 ( 1,2, , ) k s i ik m k b q b i n = = =  1 1 11 12 1 1 21 22 2 2 2 1 1 2 1 n i i i n n n i i i m m mn n n mi i i a b a a a b a a a b a b AB a a a b a b = = =                    = =                         可见，AB 的各行向量均为 1 2 , , , m m mr b b b 的线性组合，故 rank AB rank B s ( )  = 合起来即 rank AB rankA rank B ( ) min ( , )  定理：设 , m n n P A C B C     , 1 † 0 , 0 0 C      −    =   = 则 (1) (1) ( ) {1} H H A A  (2) † (1)   A A ( ){1} (3) S、T 为可逆方阵且与 A 可乘，则 1 (1) 1 ( ){1} , ( , ) m m n n T A S SAT S C T C m n − −      (4) rank ( (1) A rankA )  (5) (1) 1 AA A A A 和 （ ） 均为幂等矩阵且与 同秩 ( 2 P P= ) (6) (1) (1) (1) ( ) ( ), ( ) ( ), (( ) ) ( ) H H R AA R A N A A N A R A A R A = = = (7) (1) ( ) A A I rank A n =  = n (1) ( ) AA I rank A m =  = m (8) (1) (1) ( ) ( ) ( ) ( ) ( ) ( ) AB AB A A rank AB rank A B AB AB B rank AB rank B =  = =  = 证明：(1) (1) (1) (1) ( ) ( ) ( ) {1} H H H H H H H A A A AA A A A A = = → 