A Lie algebra view of matrix splittings

Abstract

In this paper we use some basic facts from the theory of (matrix) Lie groups and algebras to show that many of the classical matrix splittings used to construct stationary iterative methods and preconditioniers for Krylov subspace methods can be interpreted as linearizations of matrix factorizations. Moreover, we show that new matrix splittings are obtained when we specialize these splittings to some of the classical matrix groups and their Lie and Jordan algebras. As an example, we derive structured generalizations of the HSS (Hermitian and skew-Hermitian splitting) iteration, and provide sufficient conditions for their convergence.