Combinatorial characterization of the null spaces of symmetric H-matrices Original Research Article

We characterize the structure of null spaces of symmetric diagonally dominant (SDD) matrices and symmetric H-matrices with non-negative diagonal entries. We show that the structure of the null space of a symmetric SDD matrix or H-matrix A depends on the structure of the connected components of its underlying graph. Each connected component contributes at most one vector to the null space. This paper provides a combinatorial characterization of the rank of each connected component, and a combinatorial characterization of a null vector if one exists. For SDD matrices, we also present an efficient combinatorial algorithm for constructing an orthonormal basis for the null space. The paper also shows a close connection between gain graphs and H-matrices, which extends known results regarding the connection between undirected graphs and Laplacian matrices, and between signed graphs and SDD matrices. We show how to exploit these combinatorial algorithms to reliably solve certain singular linear systems in finite-precision arithmetic.