A necessary and sufficient condition for M-matrices and its relation to block LU factorization Original Research Article

We present a necessary and sufficient condition for M-matrices in terms of a special diagonal dominance. Then we use the new result to show that if the block comparison matrix of a block matrix image is an M-matrix, there exists a block permutation matrix P such that block LU factorization applied to image is stable—i.e., the norms of the block multipliers −A(k − 1)i, kA(k − 1)k, k are bounded by 1. We also present a collection of tools in the literature related to the subject matter. We define incomplete M-matrices, prove a necessary and sufficient condition for such matrices, and present their implications for block LU factorization.