I am trying to prove the following statement for a matrix $A\in\mathbb{C}^{n\times n}$
$A$ is positive definite if and only if there exists a nonsingular matrix $C\in\mathbb{C}^{n\times n}$ such that $C^*AC$ is positive definite.
In the above, the fact that $A$ is Hermitian is implicitly assumed since all positive definite matrices are Hermitian. Also the asterisk denotes the Hermitian operator.
My attempt is the following. For the forward part, assume that $A$ is positive definite. Then choose $C$ as the identity matrix, $C=I_n$. Then, indeed $C^*AC=I^*AI=A$ is positive definite by assumption.
For the reverse, assume that $C^*AC$ is positive definite for some nonsingular $C\in\mathbb{C}^{n\times n}$. Let $\mathbf{x}\in\mathbb{C}^n$ and let $\mathbf{y}=C\mathbf{x}$. By assumption, $$0<\mathbf{x}^*C^*AC\mathbf{x}=\mathbf{y}^*A\mathbf{y}$$ i.e. $A$ is positive definite.
I am not sure whether my arguments are OK.
Let $y \ne 0$, then $\exists x \ne 0$ such that $y=Cx$ since $C$ is nonsingular.
Hence $$y^*Ay=(Cx)^*A(Cx)=x^*(C^*AC)x >0$$