$\mathbf{A}$ is an $M\times N$ matrix with $M\leq N$ and $\mathbf{C}$ is an $N\times N$ diagonal matrix. $\mathbf{A}^{-1}$ does not exist, $(\mathbf{A}\mathbf{A}^H)^{-1}$ exists.
Matrix $\mathbf{A}\mathbf{C}\mathbf{A}^H$ is invertible. Is it possible to take out $\mathbf{C}^{-1}$ from $(\mathbf{A}\mathbf{C}\mathbf{A}^H)^{-1}$ , either using Kronecker operations or something else?