Question

已知实对称矩阵A=(2 -2 0 -2 1 -2 0 -2 0) 求特征向量

Answer 1

|λE-A| =
|λ-2 2 0|
|2 λ-1 2|
|0 2 λ|
|λE-A| =
|λ-2 2 0|
|2 λ-1 2|
|-λ (4+λ-λ^2)/2 0|
|λE-A| = (-2)[(λ-2)(4+λ-λ^2)/2+2λ]
= λ^3-3λ^2-6λ+8 =(λ-1)(λ+2)(λ-4)
得特征值 λ = -2， 1， 4.
对于 λ = -2，λE-A =
[-4 2 0]
[2 -3 2]
[0 2 -2]
行初等变换为
[2 -3 2]
[0 -4 4]

[0 2 -2]
行初等变换为
[2 0 -1]
[0 1 -1]

[0 0 0]
得特征向量 (1, 2, 2)^T;

对于 λ = 1，λE-A =
[-1 2 0]
[2 0 2]
[0 2 1]
行初等变换为
[-1 2 0]
[0 4 2]
[0 2 1]

行初等变换为
[-1 0 -1]
[0 2 1]
[0 0 0]

得特征向量 (2, 1, -2)^T;

对于 λ = 4，λE-A =
[2 2 0]
[2 3 2]
[0 2 4]
行初等变换为
[1 1 0]
[0 1 2]
[0 2 4]

行初等变换为
[1 0 -2]
[0 1 2]
[0 0 0]

得特征向量 (2, -2, 1)^T.