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

A proper isometry

$\varphi \colon \mathbb {R} ^{3}\longrightarrow \mathbb {R} ^{3}$

has an

eigenvector with eigenvalue

${}1$,

that is, there exists a line (through the origin) that is a fixed line for ${}\varphi$.

Proof, Write another proof