Hi, your problem is interesting. Which matrix norm do you want to minimize?

Frobenius norm. Sorry I forgot to mention that.

If there is a closed form solution I'm pretty sure it would look like A = YX^{\dagger}, where X^{\dagger} is the pseudo inverse of X. I worked through a similar proof for square matrices and a singular and symmetric X a while back. The solution in that case was unique in the Frobenius norm. Define a nonlinear functional ||Y-AX||_F^2 and take it's Gateaux derivative, see if you can work out the necessary conditions for a minimum.

Let's assume X is full rank and momentarily relax the constraint that A be full rank.  This is a linear least squares problem  (with the Frobenius norm) and it has a closed form solution.  Let C = (X^T kron I_m) where kron is the Kronecker product and I_m is an mxm identity matrix.  Then the problem is equivalent to min_z || vec(Y) - C z||_2, and it has a closed-form global, unique minimizer z=C^{dagger} vec(Y), where z = vec(A).  Uniqueness comes from X being full rank.  If that A (call it A_*) happens to be full rank, you have your solution.

If A_* isn't full rank (and X is), then there is no minimizer that is full rank because every neighborhood of A_* has a full rank A for which || Y-AX || is arbitrarily close to and greater than || Y - A_* X ||.  In other words ||Y - A_* X|| can be approached arbitrarily closely but not be achieved by full rank A (it's an infimum over full rank A and there is no minimizer).  If A_* is not full rank and X is also not full rank, then the minimizer is not unique and you can perturb A_* in the direction of the left null space of A without affecting the minimum norm.  In this case, you may be able to find a full rank A that achieves the minimum.

Thanks for your suggestion Mr. Mark Wicks. But my data matrix is very large. ie value of L is very high. When i take the kronecker product i get a huge matrix. When i try to take the psudoinverse Matlab seems to hang.

Your suggestion works well for smaller L values.

My answer is YXT(XXT)-1.