proof of Bauer-Fike theorem

We can assume λ~σ(A) (otherwise, we can choose λ=λ~ and theorem is proven, since κp(X)>1). Then (A-λ~I)-1 exists, so we can write:


since A is diagonalizable; taking the p-norm ( of both sides, we obtain:

1 = u~p
= X(D-λ~I)-1X-1rpXp(D-λ~I)-1pX-1prp
= κp(X)(D-λ~I)-1prp.

But, since (D-λ~I)-1 is a diagonal matrixMathworldPlanetmath, the p-norm is easily computed, and yields:



Title proof of Bauer-Fike theorem
Canonical name ProofOfBauerFikeTheorem
Date of creation 2013-03-22 15:33:08
Last modified on 2013-03-22 15:33:08
Owner Andrea Ambrosio (7332)
Last modified by Andrea Ambrosio (7332)
Numerical id 9
Author Andrea Ambrosio (7332)
Entry type Proof
Classification msc 15A42