On upper triangular block weak regular splittings of a singular M-matrix

AbstractLet A be in the Frobenius normal form, and let A = M − N be a weak regular splitting. We call this weak regular splitting an upper triangular block weak regular splitting if M and N are upper triangular block matrices partitioned in the same way as A. First we study weak regular splittings of a singular irreducible M-matrix and obtain some interesting results. We also investigate the index of the spectral radius of the iteration matrix M−1N when the splitting A = M − N is assumed to be upper triangular weak regular, and we obtain ind1(M−1N) ⩾ ind0(A). By this result, we improve the main results obtained by H. Schneider