Given invertible matrices $X,Y$, is there a way to find $A = A(X,Y)$ so that $AXA^T = Y$?
I am interest in the case where all the matrices are over some $\textit{finite}$ field, say $GF(2)$ for simplicity, and $Y$ is triangular (that is why the title).
I am primarily interested on finding $A$ and not its existence.