I am trying to find the matrix $M$ satisfying: $$v_t'= MA_tM^{-1}v_t$$
For a dataset of observed transformations $(v_t',A_t,v_t)_t$.
Basically I have two isomorphic vector spaces $U$ and $V$, where transformations on $U$ are known but the space itself is not observed. I am trying to describe transformations on $V$ from observing the transformations $A_t$ on the space $U$ and their effects on $V$ through observed vectors $(v_t', v_t)_t$.
For now I am solving it through Least squares: $$M = \underset{X}{argmin} \sum_t||v_t' - XA_tX^{-1}v_t||^2_2$$
But I was wondering if this is a known problem or if a closed solution exists.
I would also be interested in any factorizations of $M$ that could make it simpler, maybe orthogonal.