Eigenpairs of Normal Matrices 
Let's say $A$ is a normal matrix, that is, $AA^*=A^*A$ - then what can be said about the eigen-pairs of $A$ and $A^*$?

I'm trying to show that if $Ax = kx$, then $A^*x = \overline{k}x$. How do I proceed?
I tried the following:


*

*$Ax = kx$ implies

*$x^*A^* = \overline{k}x^*$

*Multiplying by $Ax$ on both sides and replacing $A^*A$  by $AA^*$ - but couldn't get anywhere past this.


Could someone point me in the right direction? Any help is appreciated.

Hint: We have $\|Ax-kx\|^2 =\|A^*x-\bar kx\|^2$. 
