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

Proof

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

To fact