Isometry/C/Diagonalizable/Fact

Spectral theorem for complex isometries

Let ${}V$ be a finite-dimensional ${}\mathbb {C}$-vector space, endowed with an inner product, and let

$\varphi \colon V\longrightarrow V$

be an isometry.

Then ${}V$ has an

orthonormal basis of eigenvectors

of ${}\varphi$.

In particular, ${}\varphi$ is diagonalizable.

Proof, Write another proof