A Note on the Solution of Rectangular Linear Systems by Iteration

Abstract

It is well known that the necessary and sufficient condition for a matrix to be convergent is that all of its eigenvalues in magnitude be less than unity. For linear systems with nonsingular coefficient matrix the convergence of the iterations is equivalent to the convergence of the so-called iteration matrix associated with the scheme. However, this is not the case for a linear system with singular coefficient matrix. In a paper by H. B. Keller [2] a condition on the iteration matrix is determined when the system is singular. Further, this condition is shown to be necessary and sufficient for the convergence of the iterates to a solution if it has one. In [2], however, the author considers a singular linear system with a coefficient matrix of order n. The popular concept of the Moore-Penrose generalized inverse of a matrix [3], [4] now makes it possible to extend the results of the ? 2 of [2] to include systems with rectangular coefficient matrix. The purpose of this note is to give a kind of decomposition of a coefficient matrix which enables us to carry out the aforementioned extension to rectangular systems. The development here parallels that of [2] for a square matrix. Consider a general system of equations