For a real matrix $A$ of size $n \times n$, with eigenvalues $0 < | \lambda_i | < 1$ and matrix $B$ of size $n \times m$ with $m \leq n$ of rank $m$, there exists a unique definite solution $X$ to the discrete Lyapunov equation $$ A X A^T + B B^T = X. $$ If now $AX = XA$, then $$ A A^T - I = B B^T X^{-1}. $$ Under which condition is $B^T X^{-1} = \alpha B^T$ for some scalar $\alpha$?
For $m = 1$, $B B^T$ is rank 1. As the left side is symmetric, I believe that in that case, it is always $B B^T X^{-1} = \alpha B B^T$. Is that true also for the general $m$?