Question

求相似变换矩阵P,使得|1,2,2

Answer 1

|A-λE| = (5-λ)(1+λ)^2.

所以A的特征值为 5,-1,-1

(A-5E)X = 0 的基础解系为:a1 = (1,1,1)'

(A+E)X = 0 的基础解系为:a2 = (1,-1,0)',a3 = (1,0,-1)'

将 a2,a3 正交化得 b2 = (1,-1,0)',b3 = (1/2,1/2,-1)'

单位化得 c1 = (1/√3,1/√3,1/√3)',c2 = (1/√2,-1/√2,0)',c3 = (1/√6,1/√6,-2/√6)'

令矩阵P = (c1,c2,c3),则 P^(-1)AP = ∧=diag(0,9,-1).

步骤就是这样,消化掉它,这一类题目就没问题了.