Abstract
A perturbation theory for pseudo-inverses is developed. The theory is based on a useful decomposition (theorem 2.1) ofB + -A + whereB andA arem ×n matrices. Sharp estimates of ∥B + -A +∥ are derived for unitary invariant norms whenA andB are of the same rank and ∥B -A∥ is small. Under similar conditions the perturbation of a linear systemAx=b is studied. Realistic bounds on the perturbation ofx=A + b andr=b=Ax are given. Finally it is seen thatA + andB + can be compared if and only ifR(A) andR(B) as well asR(A H) andR(B H) are in the acute case. Some theorems valid only in the acute case are also proved.
Similar content being viewed by others
References
S. N. Afriat,Orthogonal and oblique projectors and the characteristics of pairs of vector spaces, Camb. Philos. Soc., 53 (1957), 800–816.
A. Ben-Israel,On error bounds for generalized inverses, SIAM J. Numer. Anal. 3 (1966), 585–592.
Å. Björck,Solving linear least squares problems by Gram-Schmidt orthogonalization, BIT 7 (1967), 1–21.
Å. Björck,Iterative refinement of linear least squares solutions I, BIT 7 (1967), 257–278.
Ch. Davis and W. M. Kahan,The rotation of eigenvectors by a perturbation III, SIAM J. Numer. Anal. 7 (1970), 1–46.
C. A. Desoer and B. H. Whalen,A note on pseudoinverses, J. SIAM 11 (1963), 442–447.
G. H. Golub and V. Pereyra,The differentiation of pseudoinverses and nonlinear least squares problems whose variables separate, Stanford University, Computer Science Report, STAN — CS — 72 — 261 (1972).
G. H. Golub and J. H. Wilkinson,Note on the iterative refinement of least squares solution, Num. Math. 9 (1966), 139–148.
R. J. Hanson and C. L. Lawson,Extensions and applications of the Householder algorithm for solving linear least squares problems, Mathematics of Computation, Vol. 23 (1969), 787–812.
A. S. Householder,The Theory of Matrices in Numerical Analysis, Blaisdell, New York (1964).
T. Kato,Perturbation Theory for Linear Operators, Springer, Berlin (1966).
L. Mirsky,Symmetric gauge functions and unitarily invariant norms, Quart. J. Math. Oxford (2) 11 (1960), 50–59.
V. Pereyra,Stability of general systems of linear equations, aequationes mathematicae, Vol. 2 (1969), 194–206.
G. Peters and J. H. Wilkinson,The least squares problem and pseudo-inverses, The Computer Journal Vol. 13 (1970), 309–316.
A. van der Sluis,Stability of solutions of linear algebraic systems, Num. Math. 14 (1970), 246–251.
G. W. Stewart,On the continuity of the generalized inverse, SIAM J. Appl. Math., Vol. 17 (1969), 33–45.
P.-Å. Wedin,On pseudo-inverses of perturbed matrices, Lund Un. Comp. Sc. Tech. Rep. (1969).
Author information
Authors and Affiliations
Additional information
This work was sponsored in part by The Swedish Institute of Applied Mathematics.
Rights and permissions
About this article
Cite this article
Wedin, PÅ. Perturbation theory for pseudo-inverses. BIT 13, 217–232 (1973). https://doi.org/10.1007/BF01933494
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01933494