Ⅰ-A\/AA 0 A-AB 1 B 入B B入 ⅠB 两边取行列式即得 (法四) A-A B AI)0I AB AI-BA ⅠA AⅠ0 AI一ABA B入I B I 两边取行列式即得.口 法面设(4)=,则存在可阵PQ使A=P(b0)Q令QBP= C D F G 则 P-()P =AI-P-1AQ-IQBP=-(O(F D) D 00 0 Q(I-BA)Q-1=XI-QBPP-1AQ-1=AL-C D(,0 FG八(00 C 0 AI-C 0 F 0 两边取行列式即得 例9A∈Km×,B∈Km×m,且m≥n.求证: (1)AI-AB=m-nJAI- BAl (2)tr(AB)=tr(BA) (3)设B,B2,……,Bn是m个同阶矩阵A1,A2,…,Am的任何循环排列,则 A1,A2,……,Am与B1,B2,…,Bmn有相同特征多项式,因而有相同特征值和迹. 证明(1)(法一)(5)  I −A 0 λI   A λI I B  =  0 λI − AB λI λB  ,  I 0 −B λI   A λI I B  =  A λI λI − BA 0  , p JrV% 2 (5')  I A B λI   λI −A 0 I  =  λI 0 λB λI − BA   I A B λI   λI 0 −B I  =  λI − AB A 0 λI  p JrV% 2 (5<)  r(A) = r, feil P, Q,  A = P  Ir 0 0 0  Q. u QBP =  C D F G  , f P −1 (λI − AB)P = λI − P −1AQ−1QBP = λI −  Ir 0 0 0   C D F G  = λI −  C D 0 0  =  λI − C −D 0 λI  Q(λI − BA)Q−1 = λI − QBPP −1AQ−1 = λI −  C D F G   Ir 0 0 0  = λI −  C 0 F 0  =  λI − C 0 −F λI  p JrV% 2  9 A ∈ Km×n ,B ∈ Kn×m,  m ≥ n. o (1) |λI − AB| = λ m−n |λI − BA|; (2) tr(AB) = tr(BA); (3)  B1, B2, · · · , Bm  m C2adl A1, A2, · · · , Am &MNQrf A1, A2, · · · , Am ` B1, B2, · · · , Bm ^D2.m2EZ3^D2.mrLT  (1)(5R) 9