BIT Numerical Mathematics

, Volume 13, Issue 2, pp 217–232

Perturbation theory for pseudo-inverses

Authors

  • Per-Åke Wedin
    • Institute of Computer sciences
Article

DOI: 10.1007/BF01933494

Cite this article as:
Wedin, P. BIT (1973) 13: 217. doi:10.1007/BF01933494

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(AH) andR(BH) are in the acute case. Some theorems valid only in the acute case are also proved.

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© BIT Foundations 1973