Suppose matrix $A \in R^{m \times n}$, and I want to matrix factorization of $A$ to $U \in R^{m \times r}$ and $V \in R^{n \times r}$, such that $A = UV^T$.
What is the reason $r \le \min(m, n)$? It seems that when $r > \max(m, n)$, the matrix $A$ can also be recovered.
Dose the requirement of "$r$ MUST BE less and equal than $\min(m, n)$" must be followed in matrix factorization?
In an experiment I did, it seems when $r > \min(m, n)$, the results look also good.
Can somebody help me work out this question? It is confusing me.
Thanks.