Suppose $G$ is a Hermitian $n \times n$ matrix and $A$ is some $n \times n$ matrix over complex numbers. such that $G-A^H G A$ is positive-definite.
Then can we show that $G$ is invertible?
Also, can we conclude anything about the eigenvalues of the matrix $A$ in terms of the eigenvalues of $G$ (for instance, relate the number of positive eigenvalues of $G$ with the eigenvalues of $A$ of norm less than 1)?