Let M(A) denote the comparison matrix of a square H-matrix A, that is, M(A) is an M -matrix. H-matrices such that their comparison matrices are non-singular are well studied in the literature. In this paper, we study characterizations of H-matrices with singular or nonsingular comparison matrix. In particular, we analyze the case when A is irreducible and then give insights into the reducible case. The spectral radius of the Jacobi matrix of M(A) and the generalized diagonal dominance property are used in the characterizations. Finally, from these characterizations, a partition of the general H-matrix set in three classes is obtained.

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It is well known that the Schur complement of some H‐matrices is an H‐matrix. In this paper, the Schur complement of any general H‐matrix is studied. In particular, it is proved that the Schur complement, if it exists, is an H‐matrix and the class to which the Schur complement belongs is studied. In addition, results are given for singular irreducible H‐matrices and for the Schur complement of nonsingular irreducible H‐matrices. Copyright © 2009 John Wiley & Sons, Ltd.

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