Euclidean vector space/R^3/Proper isometry/Representation/Fact/Proof

Proof

Due to fact, there exists an eigenvector ${}u$ for the eigenvalue ${}1$. Let ${}U=\mathbb {R} u$ denote the corresponding line. This line is fixed and, in particular, invariant under ${}\varphi$. Because of fact, also the orthogonal complement ${}U^{\perp }$ is invariant under ${}\varphi$. That is, there exists a linear isometry

$\varphi _{2}\colon U^{\perp }\longrightarrow U^{\perp }$

that coincides on ${}U^{\perp }$ with ${}\varphi$. Here, ${}\varphi _{2}$ is proper. Therefore, by fact, ${}\varphi _{2}$ is a plane rotation. If we choose a vector of length one in ${}U$, and an orthonormal basis of ${}U^{\perp }$, then ${}\varphi$ has with respect to this basis the given matrix form.

To fact