I know that if we have a square $n \times n$ matrix $A$ in F and two vectors $v,u$ in $\mathbb F^n$, following the exchange lemma we can show that:
$$\langle Au,v\rangle = \langle u, \ A^*v\rangle $$
Suppose that I have the following dot product:
$$\langle \ A^*Av,v\rangle $$
I can show using the exchange lemma that :
$$\langle \ A^*Av,v\rangle = \langle v, \ A^*Av\rangle $$
On the other hand, I can show using the exchange lemma that:
$$\langle \ A^*Av,v\rangle = \langle Av, Av\rangle $$
Therefore I get:
$$\langle v, \ A^*Av\rangle = \langle Av, Av\rangle $$
How is it possible that $\langle v, \ A^*Av\rangle = \langle Av, Av\rangle $?