Dear  members,

I don't see how to prove the following:

Let $G$ be a compact group and  $(\pi, E)$ a finite linear representation of $G$. We consider a  a hermitian form where $<,>$ on $E$ and set
$(a,b)=\int_G <\pi_x(a),\pi_x(b)> dx$, $a,b\in G$,  where $dx$ is a  Haar measure on $G$.

The question is: Show that there exists an invertible  operator  $A:E\to E$ such that
$(a,b)=<A(a),A(b)> $ for all $a,b\in G$.

Thanks in advance,

Best regards,

Mostafa MASLOUHI.