Existence and uniqueness of splittings for stationary iterative methods with applications to alternating methods

Summary. Given a nonsingular matrix $A$, and a matrix $T$ of the same order, under certain very mild conditions, there is a unique splitting $A=B-C$, such that $T=B^{-1}C$. Moreover, all properties of the splitting are derived directly from the iteration matrix $T$. These results do not hold when the matrix $A$ is singular. In this case, given a matrix $T$ and a splitting $A=B-C$ such that $T=B^{-1}C$, there are infinitely many other splittings corresponding to the same matrices $A$ and $T$, and different splittings can have different properties. For instance, when $T$ is nonnegative, some of these splittings can be regular splittings, while others can be only weak splittings. Analogous results hold in the symmetric positive semidefinite case. Given a singular matrix $A$, not for all iteration matrices $T$ there is a splitting corresponding to them. Necessary and sufficient conditions for the existence of such splittings are examined. As an illustration of the theory developed, the convergence of certain alternating iterations is analyzed. Different cases where the matrix is monotone, singular, and positive (semi)definite are studied.