On Nonnegative Moore-Penrose Inverses of Perturbed Matrices

We consider the problem of characterizing nonnegativity of the Moore-Penrose inverse for matrix perturbations of the type A − XGY, when the Moore-Penrose inverse of A is nonnegative. Here, we say that a matrix B = (b ij ) is nonnegative and denote it by B ≥ 0 if b ij ≥ 0, ∀i, j. This problemwasmotivated by the results in [1], where the authors consider an M-matrix A and find sufficient conditions for the perturbed matrix (A − XY) to be an M-matrix. Let us recall that a matrix B = (b ij ) is said to Z-matrix if b ij ≤ 0