A Unified Proof for the Convergence of Jacobi and Gauss–Seidel Methods

We present a new unified proof for the convergence of both the Jacobi and the Gauss–Seidel methods for solving systems of linear equations under the criterion of either (a) strict diagonal dominance of the matrix, or (b) diagonal dominance and irreducibility of the matrix. These results are well known. The proof for criterion (a) makes use of Geršgorin’s theorem, while the proof for criterion (b) uses Taussky’s theorem that extends Geršgorin’s work. Hence the topic is interesting for teaching purposes.