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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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    Hint: Since $V$ has odd dimension, you know $S$ must have at least one real eigenvalue ...2012-12-02
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    You might also consider posting what work you have done so far and a bit of motivation for recent spree of linear algebra question.2012-12-02
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    Neal: and then?2012-12-02
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    "Then"? Then you're done! What are the possible eigenvalues of an orthonormal transformation? Ho are the possible eigenvalues of a power of *any* transformation related to the eignevalues of the transformation?2012-12-02
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    DonAntonio: I need more explanation here. What are the possible eigenvalues of an orthogonal transformation?2012-12-02
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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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    Why $=$, and why $S=S^T$?2012-12-02
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    @i_a_n: Check the inner product operations.2012-12-02
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    Sorry I don't think I get it. Can you explain them more clearly?2012-12-02