Classes of general H-matrices

Abstract

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.

On generalized Schur complement of nonstrictly diagonally dominant matrices and general H-matrices

“…H n , H I n , H S n and H M n will denote the set of all n × n general H−matrices, the set of all n × n invertible H−matrices, the set of all n × n singular H−matrices and the set of all n × n mixed H−matrices, respectively (see [6]). Similar to (2.1), we have…”

Schur complement of general H‐matrices

“…We remain that any matrix A ∈ H M , singular or not, has nonzero diagonal elements, its comparison matrix M(A) is singular but at least one equimodular matrix is nonsingular, see [4] for details. The matrix A can be irreducible or not, then we distinguish two cases.…”

“…From Theorem 2, S α (M(A)) is a singular and irreducible Mmatrix, and, since S α (M(A)) = M(S α (A)), S α (A) is an H-matrix not included in H I . By the irreducibility, S α (M(A)) is not in H S (see [4]). Then S α (A) remains in H M .…”

“…or nonsingularity of a principal submatrix and to the irreducibility of the original matrix. On the other hand, the partition of H-matrices according to the singularity or not of the matrix is established in reference [4], where the general set of H-matrices is split in three different classes with different properties related with the above subjects for each class of H-matrices.…”

“…See [3]). In [4] a partition of the set of all H-matrices in three classes is given: The invertible class H I containing all (nonsingular) H-matrices such that their comparison matrix is a nonsingular M -matrix; the mixed class H M formed for all H-matrices having singular comparison matrix but with at least one equimodular matrix (that is, a matrix with the same comparison matrix) being nonsingular; and the singular class H S with all (reducible) H-matrices such that all their equimodular matrices are singular.…”