Perturbation theory for pseudo-inverses

descriptionPublicationkeyboard_double_arrow_right Article 01 Jun 1973 English Publisher:Springer Science and Business Media LLCJournal:BIT, volume 13, pages 217-232 (issn: 0006-3835, eissn: 1572-9125,

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Authors: Wedin, Per-Ake;

doi: 10.1007/bf01933494

Perturbation theory for pseudo-inverses

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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.

Keywords

Linear equations (linear algebraic aspects), Numerical solutions to overdetermined systems, pseudoinverses, Theory of matrix inversion and generalized inverses

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