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取特征值矩阵 ∧ = diag(1, 1, -2), 特征向量矩阵 P = (a1, a2, a3) =
[2 -2 1]
[0 1 -1]
[1 0 -1]
(P, E) =
[2 -2 1 1 0 0]
[0 1 -1 0 1 0]
[1 0 -1 0 0 1]
初等行变换为
[1 0 -1 0 0 1]
[0 1 -1 0 1 0]
[0 -2 3 1 0 -2]
初等行变换为
[1 0 -1 0 0 1]
[0 1 -1 0 1 0]
[0 0 1 1 2 -2]
初等行变换为
[1 0 0 1 2 -1]
[0 1 0 1 3 -2]
[0 0 1 1 2 -2]
P^(-1) =
[1 2 -1]
[1 3 -2]
[1 2 -2]
则 P^(-1)AP = ∧, A = P∧P^(-1) =
[-2 -6 6]
[ 3 7 -6]
[ 3 6 -5]
[2 -2 1]
[0 1 -1]
[1 0 -1]
(P, E) =
[2 -2 1 1 0 0]
[0 1 -1 0 1 0]
[1 0 -1 0 0 1]
初等行变换为
[1 0 -1 0 0 1]
[0 1 -1 0 1 0]
[0 -2 3 1 0 -2]
初等行变换为
[1 0 -1 0 0 1]
[0 1 -1 0 1 0]
[0 0 1 1 2 -2]
初等行变换为
[1 0 0 1 2 -1]
[0 1 0 1 3 -2]
[0 0 1 1 2 -2]
P^(-1) =
[1 2 -1]
[1 3 -2]
[1 2 -2]
则 P^(-1)AP = ∧, A = P∧P^(-1) =
[-2 -6 6]
[ 3 7 -6]
[ 3 6 -5]
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