If I'm not mistaken, if a matrix $M$ has its conjugate $M^*=M$, then $M$ is Hermitian.
In this case then, am I asked to show that $(A^*A)^*=A^*A$? What does it have to do with $A$ being invertible though?
And positive definite? How do I show that it's positive definite?
Thanks!
In general, we have $(AB)^{*} = B^*A^*$. Hence, we get that $$(A^*A)^* = A^* (A^*)^* = A^*A$$ We need that $A$ is invertible to prove that $(A^*A)$ is positive definite. Consider $x \in \mathbb{C}^n$, we then have $$x^*(A^*A)x = (Ax)^*(Ax) = \Vert Ax \Vert_2^2 \color{red}{\geq} 0$$ If $A$ is invertible, then the nullspace of $A$ is trivial i.e. if $x \neq 0$, then $Ax \neq 0$. Hence, we have that if $x \neq 0$, then $$\Vert Ax \Vert_2^2 \color{blue}{>} 0$$ This shows that if $A$ is invertible, then $A^*A$ is positive definite.