I have been given a problem of solving $X$, which is an unblurred image, in the system: $$B = A X A \iff X = A^{-1} B A^{-1},$$ where matrix $A$ describes the blurring of an image and matrix $B$ is the blurred image.
Matrices $A$ and $B$ are given and it is known that $A$ is symmetrical and positive definite. This leads me to believe that Cholesky factorization is a good method to apply.
The only problem is that the assignment specifies that it wants a solution where only one factorization is used. As I see it two total factorizations are needed. One is used when solving $B = AX$ and then one more for the final matrix $A$ which then gives an unblurred image. Am I missing a way to rewrite the problem so that only factorization has to be used?