BIT Numerical Mathematics

, Volume 13, Issue 2, pp 217-232

First online:

Perturbation theory for pseudo-inverses

  • Per-Åke WedinAffiliated withInstitute of Computer sciences

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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.