Monotonicity of Nonnegative Matrices

We present a nonnegative rank factorization of a nonnegative matrix A for the case in which one or both of A(1)A and AA(1) are nonnegative. This gives, in particular, a known result for characterizing nonnegative matrices when A†A or AA† is nonnegative. We applied this characterization to the derivation of known results based on the characterization of nonnegative monotone matrices. A matrix A = (aij) is nonnegative if aij ≥ 0 for all i, j, and the nonnegativity is expressed as A ≥ 0. If there exists a matrix X such that X satisfies the following equations, for λ v {1, 2, 3, 4, 5}: (1)AXA = A, (2)XAX = X, (3)AX = (AX) , (4)XA = (XA) , and (5) AX = XA, then X is called a λ−inverse of A,, also known as a generalized inverse of A. A λ-inverse of A is denoted A. If A > 0, then A is referred to as λ-monotone. For λ = {1, 2, 3, 4}, X is the Moore–Penrose inverse of A. If λ = {1, 2, 5}, then X indicates the group inverse of A. Whereas the Moore–Penrose inverse always exists and is unique, the group inverse exists if and only if the index of A is 1 and unique. The Moore–Penrose and group inverses of A are denoted by A† and A, respectively. For λ = 1, the matrixX = A is known as the 1-inverse of A. For an example of the applications of 1-inverses in interval linear programming, see Ben–Israel and Greville [1]. Related work has bee motivated by the utility of characterizing a nonnegative matrix A such that a linear system Ax = B has a nonnegative solution or a best approximate nonnegative solution when the output matrix B is also nonnegative. Several sufficiency conditions have been demonstrated under a variety of hypotheses. For a linear system Ax = b, x = Ab is a best approximate solution to the minimum norm, or x = Ab is a solution provided that the system Ax = b is consistent. Along these lines, some authors have studied the conditions under which A is nonnegative. For example, for λ = {1, 2, 3, 4}, see ([1], Theorem 5.2), for λ = {1, 5}, see ([5], Theorem 1), and for λ = 1, see ([6], Theorem 2). Under a weaker hypotheses, Jain–Tynan [4] considered nonnegative matrices A such that AA is nonnegative or AA is nonnegative. An n × n nonnegative matrix is monomial if each row and each column has exactly one nonzero entry. Unless otherwise stated, by ”vector” we mean a ”column vector”. The purpose of this paper is to improve the known results presented in [4]. This work characterizes nonnegative matrices A such that AA or AA is nonnegative. As a consequence, some known results are obtained for the cases in which A is