Eigenpairs of Normal Matrices

Question

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.

Answer 1

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