Let $V$ be an inner product space. A linear map $U:V\rightarrow V$ is a partial isometry if there is a subspace $M\subset V$ such that $\parallel Ux\parallel =\parallel x \parallel$ for all $x\in M$ and $\parallel Ux\parallel =0$ for all $x\in M^{\perp}$.
Prove that if $\lambda$ is a eigenvalue of a partial isometry then $|\lambda|\leq 1$.
This problem was taken from Halmos's Finite Dimensional Vector Spaces (sec. 76).
Thanks.