This question is related, but distinct from, a question I posted the other day. I am looking at an iterative matrix equation $$X_{k+1} = A\,X_{k}\,A^{T} + B,$$ where $X$, $A$, and $B$ are all $n \times n$ real matrices. I was wondering if there was a "change of coordinates" to transform this equation into one of the form $$Y_{k+1} = A\,Y_{k}\,A^{T}$$ or something of the sort that was easier to analyze.
My first thought was to try something similar to the change of coordinates one would use for $$x_{k+1} = A\,x_{k} + b$$ which is $$x \to x + \left(I - A\right)^{-1}\,b$$ or the change of coordinates one would use for $$x_{k+1} = x_{k}\,A^T + b$$ which would be $$x \to x + b\,\left(I - A\right)^{-1}.$$
To this end, I considered the change of coordinates $$X \to X + C$$ where $C$ is a matrix I will determine later. Using this change of coordinates, the iterative equation becomes $$X_{k+1} = A\,X_{k}\,A^{T} + A\,C\,A^{T} - C + B.$$ So $C$ would have to satisfy $$B = C - A\,C\,A^{T}.$$ The difficulty here is that $C$ is being multiplied on the left and right by a matrix, so it isn't apparent to me how to combine the terms on the right.