matrix resolvent properties

The matrix resolvent norm for a complex-valued s is related to the proximity of such value to the spectrum of A; more precisely, the following simple yet meaningful property holds:


where . is any self consistent matrix normMathworldPlanetmath, σA is the spectrum of A and the distance between a complex point and the discrete set of the eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath λi is defined as dist(s,σA)=min1in|s-λi|.

From this fact it comes immediately, for any 1in,



Let (λi,𝐯) be an eigenvalue-eigenvector pair of A; then


which shows (s-λi) to be an eigenvalue of (sI-A); (s-λi)-1 is then an eigenvalue of (sI-A)-1 and , since for any self consistent norm |λ|A, we have:




Title matrix resolvent properties
Canonical name MatrixResolventProperties
Date of creation 2013-03-22 15:33:52
Last modified on 2013-03-22 15:33:52
Owner Andrea Ambrosio (7332)
Last modified by Andrea Ambrosio (7332)
Numerical id 15
Author Andrea Ambrosio (7332)
Entry type Result
Classification msc 15A15