I would like to do non negative mf and I wanna ask a question about my main matrix. The question is should I include rows and columns that have no non-zero entry in them. I think if there is not a single relation information between any row and column, it doesn't make sense. Is that correct to prune those rows and columns that don't have a single non-empty entry from the main matrix?
No. If you have a zero row but a nonzero corresponding right hand side, then the linear system has no solution. If you have a zero column, then the solution has infinitely many solutions because the entry in the corresponding position of the solution vector does not matter -- any value there is allowed, and so the solution is not unique.
In other words, empty rows or columns of a matrix have meaning.