线性代数题，如图，请问怎么证明A的任一特征向量均能由ξ线性表出？

Question

如题所述

Answer 1

Aξ = λξ, 得
a-3 = λ
b+2 = λ
1 = -λ
解得 λ = -1， a = 2， b = -3
|λE-A| =
|λ-2 1 -2|
| -5 λ+3 -3|
| 1 0 λ+2|
|λE-A| =
|λ-2 1 2-λ^2|
| -5 λ+3 7+5λ|
| 1 0 0|
= λ^3+3λ^2+3λ+1 = (λ+1)^3
λ = -1, -1, -1
(λE-A) =
[-3 1 -2]
[-5 2 -3]
[ 1 0 1]
初等行变换为
[ 1 0 1]
[ 0 1 1]
[ 0 0 0]
得特征向量为 k(1 1 -1)^T
即 kξ本回答被网友采纳