实反对称矩阵的特征值只能为零或纯虚数怎么证?实反对称矩阵的特征值只能为零或纯虚数怎么证明啊?

Proof:Suppose A is a reel skew-symmetric matrix,and λ is a eigenvalue of A.
That is,Aα=λα （α=(a1,a2,...,an)'）
we multply by （α共轭）’on both sides
(α共轭)'Aα=(α共轭)'λα=λ(α共轭)'α
on the other hand
(α共轭)'Aα=(α共轭)'(-A')α=-(Aα的共轭)'α=-(λα共轭)'α
so λ（α共轭）'α=-(λα共轭)'α=-λ（α共轭）'α
so λ=-λ
we suppose λ=a+bi
that is a=0
λ=0 or λ=bi