Euclidean vector space/R^3/Isometry/Eigenvector/Fact/Proof

Proof

The characteristic polynomial ${}P$ of ${}\varphi$ is a normed polynomial of degree three. For ${}t\to +\infty$, we have ${}P(t)\to +\infty$, and for ${}t\to -\infty$, we have ${}P(t)\to -\infty$. Due to the intermediate value theorem, ${}P$ has a real zero. Such a zero is an eigenvalue of ${}\varphi$. Because of fact, this eigenvalue equals ${}1$ or ${}-1$.

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