  It is well known that as a famous type of iterative methods in numerical
linear algebra, Gauss-Seidel iterative methods are convergent for linear
systems with strictly or irreducibly diagonally dominant matrices, invertible
$H-$matrices (generalized strictly diagonally dominant matrices) and Hermitian
positive definite matrices. But, the same is not necessarily true for linear
systems with nonstrictly diagonally dominant matrices and general $H-$matrices.
This paper firstly proposes some necessary and sufficient conditions for
convergence on Gauss-Seidel iterative methods to establish several new
theoretical results on linear systems with nonstrictly diagonally dominant
matrices and general $H-$matrices. Then, the convergence results on
preconditioned Gauss-Seidel (PGS) iterative methods for general $H-$matrices
are presented. Finally, some numerical examples are given to demonstrate the
results obtained in this paper.
