Convergent Powers of a Matrix with Applications to Iterative Methods for Singular Linear Systems

For a square, possibly singular, matrix A decomposed as $A = M - N$ where M is nonsingular, let T = M- 1 N. The Drazin inverse of $I - T$ is used to review well-known conditions under which the powers of T converge to some matrix. These concepts are then applied to the study of the convergence of the linear stationary iterative process x(k + 1) = Tx⁽ᵏ⁾ + M- 1 b, which is used to approximate solutions to consistent linear systems $Ax = b$ . When the process converges, the limit is given in terms of the Drazin inverse of $I - T$ and asymptotic rates of convergence are discussed. The concept of a regular splitting of a nonsingular matrix is extended to the singular case in a natural way and convergence criteria are established. Finally, it is shown that a matrix A has a regular splitting $A = M - N$ such that the powers of T = M- 1 N converge if and only if $A = AXA$ is solvable for some nonsingular X 0, thus providing a complete extension of Varga’s characterization of a convergent regular splitting to the general case.