  We analyze the convergence of the (algebraic) multiplicative Schwarz method
applied to linear algebraic systems with matrices having a special block
structure that arises, for example, when a (partial) differential equation is
posed and discretized on a domain that consists of two subdomains with an
overlap. This is a basic situation in the context of domain decomposition
methods. Our analysis is based on the algebraic structure of the Schwarz
iteration matrices, and we derive error bounds that are based on the block
diagonal dominance of the given system matrix. Our analysis does not assume
that the system matrix is symmetric (positive definite), or has the $M$- or
$H$-matrix property. Our approach is motivated by and significantly generalizes
an analysis for a special one-dimensional model problem given in [4].
