When are nonnegative matrices product of nonnegative idempotent matrices?

Applications of nonnegative matrices have been of immense interest to both social and physical scientists, particularly to economists and statisticians. This paper considers the question as to when a nonnegative singular matrix can be decomposed as a product of nonnegative idempotents analogous to the well-known result for any arbitrary matrix. It is shown that (i) all singular nonnegative matrices of rank <3, (ii) all singular nonnegative matrices that have a nonnegative von Neumann inverse, (iii) (0-1) nonnegative definite matrices and (iv) periodic matrices have the property that they decompose into a product of nonnegative idempotents. An example is given, in general, showing that this need not be true for singular matrices of rank three or higher, including stochastic and symmetric matrices. Besides computational techniques, a recent result that a singular nonnegative quasi-permutation matrix is a product of nonnegative idempotents plays a key role in the proofs of the results.