Let $V$ be a real inner product space of odd dimension and $SāL(V,V)$ an orthogonal transformation. Prove that there is a vector $v$ such that $S^2(v)=v$.
proof about orthogonal transformation in an inner product space
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linear-algebra
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0Actually I don't understand why $S$ must have at least one real eigenvalue ... ā 2012-12-02
1 Answers
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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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0Sorry I don't think I get it. Can you explain them more clearly? ā 2012-12-02