Finding mapping between two spatial representations of the same objects I have two matrices $U$ and $V$.  $U$ is $n \times n$ and $V$ is $n \times m$. (Both are empirical results of an experiment.) I would like to find a linear transformation $A$, $m \times n$, such that $VA = \hat{U}$, where $\hat{U} = \underset{X}{\arg \min}$ $m(X, U)$ for some metric $m$.

Is there a closed form or iterative solution to this problem?
If your error norm is the Frobenius distance (that is, $m(X, U) = \|X - U \|_F$, then there is a closed form for your problem. $A = V^+ U$, where $V^+$ is the Moore-Penrose pseudoinverse of $V$ (this is a generalized inverse for non-square matrices).