I tend to think the statement is true and is provable, mainly because I've tried coming up with various disproofs yet all seem to work.
A matrix $A\in M_n(\mathbb{C})$ is normal if and only if for all $v\in\mathbb{C^n},||Av||=||A^*v||$
I'm aware that becasue $A\in M_n(\mathbb{C})$ is normal, than $A^*A=AA^*$
Proving the $\rightarrow$ part of the iff went out like so: $$ ||Av||=\sqrt{<Av,Av>}=\sqrt{A\cdot A^*<v,v>}=\sqrt{A^*\cdot A<v,v>}=\sqrt{<A^*v,A^*v>}=||A^*v|| $$ yet when I got to the $\leftarrow$ part, I ran into some thoughts weather this statement is even correct.
Any thoughts would be helpful.
Thanks!
We have $\lVert Av \rVert = \lVert A^*v \rVert$ for every $v \in \mathbb{C}$. Thus, by squaring both sides we get $$ <Av, Av> = <A^*v, A^*v> \iff \\<v, A^*Av> = <v, AA^*v> \iff \\ <v, (A^*A-AA^*)v> = 0$$ This last identity holds for every $v \in \mathbb{C}$ if and only if $A^*A - AA^* = 0$, i.e. $A$ is normal.