Approximate solution to a matrix equation

Question

Let $A$ and $B$ be $n \times m$ matrices. I am looking for a $m \times m$ matrix $X$ which would be an approximate solution to the equation $AX = B$ (an exact solution is very unlikely to exist). More specifically, the following Frobenius norm: $\| AX-B \|$ should be minimized.

What are the plausible algorithms/methods for this problem?

Answer 1

This question is a standard one and the tensor product has nothing to do here. According to Gauss, we must solve the equation in $X$: $(A^TA)X=A^TB$. If $rank(A)=m\leq n$, then there is a sole solution $X=(A^TA)^{-1}(A^TB)$.

Answer 2

You write this using kronecker products and vectorization as $ (I_m \oplus A) \text{vec}(X) = \text{vec}(B)$. This is a standard linear systems of equations, and the solution you are looking for is simply the least-squares solution. In MATLAB

kron(eye(m),A)\B(:)