If $A$ is such that if $u$ and $v$ are orthogonal than $A(u)$ and $A(v)$ are also orthogonal (that is, their scalar multiple is $0$) for any $(u,v)$ pair and $A \in Hom (V,V)$ where V is a complex vector space 

then prove that A is a scalar (real,non-negative) multiple of a unitary transformation(so $A=\lambda D$ where $D^\*D=I$ and $\lambda \ge0$ and $\lambda \in \mathbb R$).