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Let $V$ be a real inner product space of odd dimension and $S∈L(V,V)$ an orthogonal transformation. Prove that there is a vector $v$ such that $S^2(v)=v$.

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    Actually I don't understand why $S$ must have at least one real eigenvalue ... – 2012-12-02

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Hint: Orthogonal transformation in inner product spaces satisfies the following relation

$ = = ,$

and have the property $S=S^T$. Note that, $=u^T \, v$.

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    Sorry I don't think I get it. Can you explain them more clearly? – 2012-12-02