(0 )2 1436587@9BAC9EDGF4HI 7P5QHRFTSVU0DXW6F4H

114 CAPÍTULO 10. ¢ § ¢ ¢¥£ £ ¢£ ¤ ¢¥£ ¡¤¥¡ ¢¡¤¥
n ∈ IN
(P −1AP ) 2n =
(P −1 AP ) 2 n
= (I3) n = I3
(P −1 AP ) 2n+1 = (P −1 AP ) 2n (P −1 AP ) = P −1 AP
(P −1AP ) 2 = (P −1AP ) (P −1AP ) = P −1A (P P −1 ) AP
= P −1 AI2AP = P −1 AAP = P −1 A 2 P
n ∈ IN
¢
−1 n −1 n
P AP = P A P
−1 2n −1 2n (P AP ) = P A P = I3
=⇒
⎧⎪ ⎨
(P −1AP ) 2n+1 = P −1A2n+1P = P −1AP
2n −1 A = P P = I3
A 2n+1 = P (P −1 AP ) P −1 = A
⎪⎩
(P −1 AP ) 12 = I3
(P −1 AP ) 9 = P −1 AP
A 12 = I3
A 9 = A.
¡ ¨ £¢
A =
¤
1 0 −1
1 2 1
2 2 3
B =
0 1 0
0 0 1
1 −3 3
¡ ¢ ¨ ¢ A ¡ ¢ ¨
B