Existence and Uniqueness of Splittings for Stationary Iterative Methods with Applications to Alternating Methods

Given a nonsingular matrix A, and a nonnegative matrix T , 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. It is shown that given a nonnegative matrix T and a splitting A = B C such that T = B 1 C, there are in nitely many other splittings corresponding to the same matrices A and T . Furthermore, some of these splittings can be regular splittings, while others can be only weak splittings. Analogous results hold in the symmetric positive semide nite case. Given a singular matrix A, not for all iteration matrices T there is a splitting corresponding to them. Necessary and su cient conditions for the existence of such splittings are given. As an illustration of the theory developed, the convergence of certain alternating iterations is analyzed. Di erent cases where the matrix is monotone, singular, and positive (semi)de nite are studied.