How to prove that the hermitian matrix $AA^\dagger$ is nonnegative?

Question

If $A$ is a square matrix and $x$ is a vector, I want to prove that the real number $\Re{(x^{\dagger}AA^\dagger x)}$ is nonnegative. In terms of components I arrived at $\Re(\sum_{i}\sum_{j}\sum_{k}\overline{x_i}A_{ik}\overline{A}_{jk}x_j)$. If $i=j$, this real number is nonnegative, but what can I say when $i\neq j$ ?

Answer 1

Write $y=A^\dagger x$. Then $x^\dagger AA^\dagger x=y^\dagger y$. But if $y_1,\ldots,y_n$ are the entries of $y$ then $y^\dagger y =|y_1|^2+\cdots+|y_n|^2$.