2-norm of product of normal matrix and vector

Question

$A \in M_n(\mathbb{C})$ is normal if $AA^*=A^*A$. Let $A \in M_n(\mathbb{C})$ be a normal matrix and $X\in \mathbb{C}^n$, we define $Y=AX \in \mathbb{C}^n$ and $Z=A^*X \in \mathbb{C}^n$. Prove that $||Y||_2= ||Z||_2$.

$AA^*=A^*A \Rightarrow AA^*X=A^*AX \Rightarrow AZ = A^*Y \Rightarrow ||AZ||_2=||A^*Y||_2$

I don' t know how to continue from here. Any hints?

Answer 1

\begin{align}{\|Y\|_2}^2&=Y^*.Y\\&=(AX)^*.AX\\&=X^*A^*AX\\&=X^*AA^*X\\&=(A^*X)^*.A^*X\\&=Z^*.Z\\&={\|Z\|_2}^2.\end{align}