This paper determines the nonnegativity of the principal components of an n x n nonnegative matrix P in terms of the marked reduced graph R(A) of A = P - rho(P)I, the minus M matrix which can be associated with P. We then apply this result to consider various types of nonnegative bases for the Perron eigenspace of P which can be extracted from a certain nonnegative matrix which is a polynomial in P. We also obtain a characterization for the eigenprojection on the Perron eigenspace of P to be, itself, a nonnegative matrix. Our results provide new proofs and extensions of results of Friedland and Schneider and of Hartwig, Neumann, and Rose.
