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-1N. 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 line...

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Bibliographic Details
Published in:SIAM journal on numerical analysis 1977-09, Vol.14 (4), p.699-705
Main Authors: Meyer, Carl D., Plemmons, R. J.
Format: Article
Language:English
Subjects:
Decomposition
Eigenvalues
Iterative methods
Iterative solutions
Linear equations
Linear systems
Mathematics
Matrices
Terminology
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Summary:For a square, possibly singular, matrix A decomposed as A = M - N where M is nonsingular, let T = M-1N. 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-1b, 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-1N 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.
ISSN:0036-1429
1095-7170
DOI:10.1137/0714047