Q:

Question about normal matrix

Let $A\in M_n$ be a normal matrix. Prove that $Ax=0$ if and only if $A^*x=0$
and the star is for complex conjugate.
I am not sure how I would do this one the definition of normal matrix is $AA^{*}=A^{*}A$
But how would I start.

A:

Hint:
Let's try the following:
$$\left\|A^*x\right\|^2=\langle A^*x,A^*x\rangle=\langle x, AA^*x\rangle=\langle x,A^*Ax\rangle=\langle Ax,Ax\rangle=\left\|Ax\right\|^2=0$$
Complete and fill in details

