Let $A$ be a normal matrix and $\lambda$ a scalar.
Show that $A-\lambda I$ is also a normal matrix.
$A$ is a normal matrix then there is a unitary diagonalization of $A$ over $\mathbb{C}$. This means that there is $P$ and $D$ such as $P^*AP=D$.
I'm not sure how to continue from here.
Thank you in advance.
Hint
Notice that $(A-\lambda I)^*=A^*-\overline \lambda I$ and that $I$ commutes with all matrices.