Reading this paper and trying to get a gasp of it, in the introduction section it talks about a linear system of equations which is $Y=DX$ where $D \in \mathbb R^{n\times K}$. The part I am trying to understand is as follows:
if $n<K$ and $D$ is a full rank matrix, an infinite number of solutions are available for the problem, hence constraints on the solution must be set.
What I understand from this answer is quite different from what they are claiming to be the case. My questions are:
- What is the condition $D$ being full rank doing? Why is it needed? Since we already know that's an underdetermined system and it has infinitely many solutions.
- Then it says we need to set constraints, but the answer I linked to states that we more constraints we will have more equations and the case would turn into a different one. Is it right?
This equation has K unknowns and n conditions. Because $n \lt K$, it needs more constraints in order to have an unique solution. Because D is full rank, so likely, it needs another $K-n$ independent constraints.