Let $A$ any matrix so $A^H A$ is Hermitian and $A^H A $ is positive definite.

Question

Question:
Let $A$ any matrix so $A^H A$ is Hermitian and $A^H A $ is positive definite.

Answer:
The first part is easy as: $A^H A =A A^H \Rightarrow AA^H = (A^HA)^H=(AA^H)^H$.
But how to prove (quickly) $<u|AA^H|u> \geq 0$?

Answer 1

$u^{H}A^{H}Au$ = $(Au)^{H}(Au)=||Au||^{2}$