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 $S^2(v)=v$.
proof about orthogonal transformation in an inner product space
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linear-algebra
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1Hint: Since $V$ has odd dimension, you know $S$ must have at least one real eigenvalue ... – 2012-12-02
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0You 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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1Neal: and then? – 2012-12-02
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0"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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0DonAntonio: I need more explanation here. What are the possible eigenvalues of an orthogonal transformation? – 2012-12-02
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0Actually I don't understand why $S$ must have at least one real eigenvalue ... – 2012-12-02