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

A proper isometry

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

has an

eigenvector with eigenvalue

${\displaystyle {}1}$,

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