How to prove that the linear rank of a matrix $\le$ its nonnegative rank?

Question

I am studying nonnegative matrix factorization and came across this proposition in some lecture notes. In this context the nonnegative rank of a matrix $A$ is, I believe, the smallest number $r$ such that $A = \sum_{i=1}^r v_i w_i'$ where $v \ge 0, w \ge 0$ .

Edit: correcting the definition of nonnegative rank (h/t @angryavian)

Answer 1

I'm not sure your definition is the correct definition of nonnegative rank. (Shouldn't $v_i' w_i$ be the outer product $v_i w_i'$? And I think the condition is that $v_i w_i'$ needs to have nonnegative entries, not $\langle v,w\rangle > 0$.)

According to Wikipedia, the nonnegative rank is the smallest $r$ for which the matrix can be written as a sum of $r$ nonnegative matrices of rank $1$.

Since the usual rank is the smallest $r$ for which the matrix can be written as a sum of $r$ [not necessarily nonnegative] matrices of rank $1$, we immediately have the desired inequality.