Question

求大神帮忙做下这道线性代数题

Answer 1

|λE-A| =
|λ-1 2 -2|
|2 λ-1 -2|
|-2 -2 λ-1|
= (λ-1)^3+8+8 - 4(λ-1) - 4(λ-1) - 4(λ-1)
= λ^3-3λ^2-9λ+27 =(λ+3)(λ-3)^2
特征值 λ = 3，3，-3。
对于 λ = 3， λE-A =
[ 2 2 -2]
[ 2 2 -2]
[-2 -2 2]
初等行变换为
[ 1 1 -1]
[ 0 0 0]
[ 0 0 0]
得特征向量 (1 -1 0)^T, (1 0 1)^T
对于 λ = -3， λE-A =
[-4 2 -2]
[ 2 -4 -2]
[-2 -2 -4]
初等行变换为
[ 2 -4 -2]
[ 0 -6 -6]
[ 0 -6 -6]
初等行变换为
[ 1 -2 -1]
[ 0 1 1]
[ 0 0 0]
初等行变换为
[ 1 0 1]
[ 0 1 1]
[ 0 0 0]
得特征向量 (1 1 -1)^T,
记 ∧ = diag(3， 3， -3), P =
[ 1 1 1]
[-1 0 1]
[ 0 1 -1]
则 ∧^2 = 3^2 E, P^(-1)AP = ∧, A = P∧P^(-1)
A^8 = P∧P^(-1)P∧P^(-1)P∧P^(-1)......P∧P^(-1)P∧P^(-1)
= P∧^8P^(-1) = P(3^8E)P^(-1) = 3^8E
f(A) = A^8 - 6400E = (3^8 - 6400)E = 161E