|A-λE| =
-1-λ 3 3
3 -1-λ 3
3 3 -1-λ
=
5-λ 3 3
5-λ -1-λ 3
5-λ 3 -1-λ
=
5-λ 3 3
0 -4-λ 0
0 0 -4-λ
= (5-λ)(-4-λ)^2.
A的特征值为5,-4,-4
(A-5E)X=0 的基础解系为:a1=(1,1,1)^T
(A+4E)X=0 的基础解系为:a2=(1,-1,0)^T,a3=(1,0,-1)^T
令P=(a1,a2,a3),则P可逆,且 P^-1AP=diag(5,-4,-4).