Let $\alpha: V \rightarrow V$ be a linear transformation defined on a finite dimensional inner product space $V$ over $\mathbb{C}$. Recall that $\alpha$ is normal if $\alpha$ and its adjoint $\alpha^{*}$ commute. Show that $\alpha$ being normal is equivalent to each of the following statements:

(i) $\alpha=\alpha_{1}+i \alpha_{2}$ where $\alpha_{1}, \alpha_{2}$ are self-adjoint operators and $\alpha_{1} \alpha_{2}=\alpha_{2} \alpha_{1}$;

(ii) there is an orthonormal basis for $V$ consisting of eigenvectors of $\alpha$;

(iii) there is a polynomial $g$ with complex coefficients such that $\alpha^{*}=g(\alpha)$.