T is a linear operator on a finite-dimensional inner-product space. Prove that $T$ is normal iff for every vector $a$ and scalar $c$ that $Ta = ca$ we also have $T^*a = \overline{c}a$.
So I've done one side of the proof but I'm stuck doing the other side; assume $TT^* = T^*T$ then we have:
$ (TT^* a \mid a) = (T^*T a \mid a) \implies (T^*a \mid T^*a) = (Ta \mid Ta) \implies \|Ta\| = \|T^*a\| $
we also have $(T - cI)^* = T^* - \overline{c}I$ which easily implies $(T - cI)$ is normal. adding these together we conclude:
$\|(T - cI)a\| = \|(T - \overline{c}I)a\|$
which concludes one side of the theorem.