Is the matrix $(A - D^{T}AD)$ positive definite if the spectral radius of $D$, $\rho(D)<1$? Here $A$ is a positive definite matrix and $D^{T}$ denotes the transpose of $D$. $A$ and $D$ are both real matrices. 

My intuition says so but I am not being able to prove rigorously. One can assume $D^{T}AD = DAD^{T}$, if necessary.