BLOCK LU FACTORIZATIONS OF M-MATRICES

It is well known that any nonsingular M-matrix admits an LU factorizat ion into M-matrices (with L and U lower and upper triangular respectiv ely) and any singular M-matrix is permutation similar to an M-matrix w hich admits an LU factorization into M-matrices. Varga and Cai establi sh necessary and sufficient conditions for a singular M-matrix (withou t permutation) to allow an LU factorization with L nonsingular. We gen eralize these results in two directions. First, we find necessary and sufficient conditions for the existence of an LU factorization of a si ngular M-marix where L and U are both permitted to be singular. Second , we establish the minimal block structure that a block LU factorizati on of a singular M-matrix can have when L and U are M-matrices.