The nonnegative rank factorizations of nonnegative matrices

Abstract

Let A ∈ P m × n r , the set of all m × n nonnegative matrices having the same rank r. For matrices A in P m × n n , we introduce the concepts of “ A has only trivial nonnegative rank factorizations” and “ A can have nontrivial nonnegative rank factorizations.” Correspondingly, the set P m × n n is divided into two disjoint subsets P (1) and P (2) such that P (1)∪ P (2) = P m × n n . It happens that the concept of “ A has only trivial nonnegative rank factorizations” is a generalization of “ A is prime in P n × n n .” We characterize the sets P (1) and P (2). Some of our results generalize some theorems in the paper of Daniel J. Richman and Hans Schneider [9].