Finding basis functions

Question

I have a matrix vector equation of the form $$\mathbf{y}=A\mathbf{x}$$ where $ \mathbf{y}$ is an $M \times 1$ vector which is known and $\mathbf{x}$ is an $N \times 1$ vector which is also known and $M >> N$ (Overdetermined system). I would like to know the columns of $A$ which are the unknown. Are there ways to find them other than by least squares strategy? Thanks.

Answer 1

There is no unique solution to this problem. You've got $M$ equations and $M\times N$ unknowns (the coefficients of $A$). Which solution would you want?

Since it's a linear problem, you could rewrite it in the more conventional form $Ba=b$ where the vector $a$ stacks the $M\times N$ coefficients of $A$ and $B, b$ known.