Let us assume $A$,$B$ and $C$ are known affine transformation matrices in homogeneous 2D space.
If it should happen that $C=A^m B^n$ for some unknown $m,n$, is there a way to detect this short of trying all the combinations? If one of $A,B$ is an identity matrix this can be determined by examining the eigenvectors, but is there an equivalent way of doing this for a product such as this?
The use-case is that I am examining a grid-like structure generated by composing transformations as described above, and I want to know if an arbitrary transformation I get can in fact be expressed as a product of exponentiated "primitive" transformations that I already know.
If $A$ and $B$ commute, you can use the matrix logarithm! Sure, it's not unique but solutions only vary by integer multiples of $2\pi i$ you can check by hand whether $\log C$ looks like $$ \log(C) = q2\pi i I + n\log(A) + m\log(A) $$
for integers $q,n,m$. Each entry in $\log(C)$ corresponds to a linear equation $q,n,m$ must solve. It's complex, but that doesn't matter. It's now a linear system.
Here's why commuting matters. In the real setting, $\log(ab) = \log(a)+\log(b)$ is always true. However, in the matrix setting addition does commute but multiplication may not. It turns out that $$ \log(AB) = \log(A)+\log(B) \quad\text{mod}\;2\pi i I $$ is only guaranteed for $A$ and $B$ which commute. So while you can do the computation described above for $A$ and $B$ which don't commute, it'll just give you garbage. The equation $C=A^nB^m$ need not have any relation with $\log(C) = q2\pi i I + n\log(A) + m\log(A)$.
Computing logarithms is not too hard numerically. Just use the Taylor series. Computing them by hand is not too bad if $A$ or $B$ is diagonalizable.
If you go the numeric route, just be sure to check that the candidate $n$ and $m$ actually work. The matrix log can be a little sensitive near the edges of it's interval of convergence.
I know that this $A$ and $B$ commuting is a fairly special case, but you gotta start somewhere. I'll keep thinking of what to do in the more general case.