Unitary vector space/Isometry/Eigenvalues/Fact

Let $V$ be a finite-dimensional $𝕂$-vector space, and let

$\phi :V⟶V$

denote a linear isometry.

Then the

modulus of every eigenvalue

of $\phi$ is $1$.

In case $𝕂=ℝ$, only the eigenvalues $1$ and $-1$ are possible.

Proof, Write another proof