Question

这题怎么做的，线性代数，谢谢大神

Answer 1

|λE - A| =
|λ-1 -2 -3|
|-2 λ-1 -3|
|-3 -3 λ-6|
= (λ-6) (λ-1)^2 - 18 - 18 - 9 (λ-1) - 9 (λ-1) - 4 (λ-6)
= λ^3 - 8λ^2 - 9λ = λ(λ+1) (λ-9)
得特征值 λ = 9， 0， -1.
对于λ = 9，λE - A = 9E - A =
[ 8 -2 -3]
[-2 8 -3]
[-3 -3 3]
初等行变换为
[1 1 -1]
[0 -10 5]
[0 10 -5]
初等行变换为
[1 1 -1]
[0 2 -1]
[0 0 0]
得特征向量 (1, 1, 2)^T.
对于λ = 0，λE - A = - A =
[-1 -2 -3]
[-2 -1 -3]
[-3 -3 -6]
初等行变换为
[1 2 3]
[0 3 3]
[0 3 3]
初等行变换为
[1 0 1]
[0 1 1]
[0 0 0]
得特征向量 (1, 1, -1)^T.
对于λ = -1，λE - A = -E - A =
[ -2 -2 -3]
[-2 -2 -3]
[-3 -3 -7]
初等行变换为
[2 2 3]
[6 6 14]
[0 0 0]
初等行变换为
[2 2 3]
[0 0 5]
[0 0 0]
得特征向量 (1, -1, 0)^T.