how to prove positive definite complex matrix

Question

In order to prove that that a quadratic function is convex: $x^HAx$ , $A$ needs to be positive semi definite. Where $A= \Phi^H \Phi$ and $\Phi$ is a n×k matrix with possibly k linearly independent columns. How to prove that A is positive semi definite?

Answer 1

$A^*A=B$ if and only if $B$ is positive semi-definite.

$\langle A^*Ax,x\rangle=\langle Ax,Ax\rangle=||Ax||^2\ge 0$ for all $x\in {\mathbb{C}}^n$

Answer 2

Note that $$\overline{z} z\geq 0$$ for any complex number $z.$

$$x^H A x= x^H{\Phi^H}\Phi x = {(\Phi x)}^H {\Phi x}=\overline{\Phi x} \Phi x.$$