A converse of schur's lemma Suppose $\rho: G \rightarrow GL(V)$ is a representation. and if $T: V \rightarrow V$ is a linear operator such that $T\circ \rho_g= \rho_g\circ T$ for all $g\in G$ implies $T=k\cdot Id$ for some number $k$. (i.e. $T$ is $G$-invariant/ $G$-intertwining implies $T$ is a homothety). Prove that $\rho$ is irreducible.
Attempt: Suppose $W$ is a vector subspace of $V$ such that $\rho_g (W) \subseteq W$ for all $g\in G$, I want to prove that $W=\{0\}$ or $V$. The only theorem I have learnt to prove irreducibility is $\langle \chi,\chi \rangle=1$ but I dont think it is useful here. But then I have no idea how to make use of the condition "the only $G$-interwining linear operator is homothety".  Please helps

Hint: If there is a non-trivial proper subspace, find a $G$-complement. Think about the projection map to one of the factors. 
