Unitary vector space/Isometry/Orthogonal complement/Fact

Let ${}\varphi \colon V\rightarrow V$ be a linear isometry on a finite-dimensional ${}{\mathbb {K} }$-vector space ${}V$, endowed with an inner product. Let ${}U\subseteq V$ denote an invariant linear subspace.

Then also the

orthogonal complement

${}U^{\perp }$ is invariant.

In particular, ${}\varphi$ can be written as a direct sum

${}\varphi =\varphi _{U}\oplus \varphi _{U^{\perp }}\,,$

where the restrictions ${}\varphi _{U}$ and ${}\varphi _{U^{\perp }}$ are also isometries.

Proof, Write another proof