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

摘要

For a square, possibly singular, matrix A decomposed as A=M-N where M is nonsingular, let T=(1/M)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 superscript(k+1)=Tx superscript(k) + (1/M)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 spliting 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=(1/M)N converge if and only if A=AXA is solvable for some nonsingular X > or = 0, thus providing a complete extension of Varga's characterization of a convergent regular splitting to the general case.