Let $V$ be a finite-dimensional inner product space over $\mathbb{C}$ and $T: V \to V$ a linear transformation.
Show that if every eigenvector of $T$ is also an eigenvector of $T^{*}$ then $T$ is a normal.
We need to show that $\forall v \in V,\ TT^*v=T^*Tv$.
I started by picking $v$ to be an eigenvector of $T$, hence $Tv=\lambda_1 v$, and $T^*v=\lambda_2 v$. Therefore: $TT^*v=T(\lambda_2 v)=\lambda_2 Tv=\lambda_1 \lambda_2 v=...=T^*Tv$
But I'm stuck when $v$ is not an eigenvector.
(Also, this question is taken from a chapter about Jordan forms, but I can't see how it relates)