Suppose $A \in M_{n\times n}(\mathbb C)$ and let $B=A A^*$.
Show that all the eigenvalues of $B$ are non-negative real.
Can you please give me an hint how to start the proof?
All I know is that $B$ is a product of the matrix $A$ with its conjugate and I can't see how it helps me.
The definition of the adjoint says that:
$$\langle x,Ay \rangle = \langle A^*x,y \rangle$$
if $A^*$ is the adjoint of $A$ with respect to $\langle \cdot,\cdot \rangle$ and $x,y$ are vectors. We can apply this here to get:
$$\langle x,A A^* x \rangle = \langle A^* x,A^* x \rangle = \| A^* x \|^2.$$
Now consider what happens if $A A^* x = \lambda x$.