Problem:
$W$ is a complex inner product space. $A$ is a linear map defined on $W$ that satisfies: $\left \langle x,y \right \rangle=0\Rightarrow \left \langle Ax,Ay \right \rangle=0$ for any $x,y\in W$. The question of the problem is to prove that $A=\lambda B$, where $B$ is unitary.
I used the hint provided by the book, to reach the point: $A^{*}A=\lambda I$. If $\lambda=0$, then $A=0$, so $A=0.B$ where $B$ is any unitary matrix of the same size as $A$. Well, for $\lambda \neq 0$, then how do prove that there exits $B$ unitary such that $A=\lambda B$?