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Pertubation theory for eigenvalues of matrices

Matthias Engelmann
Foredrag for studerende
Fredag, 7 februar, 2014, at 15:15-16:00, in Aud. D4 (1531-219)
Abstrakt:
In this very basic lecture give an introduction to the theory of perturbation of point spectra. Starting with a self-adjoint, linear map $T$ on $\mathbb{C}^n$ I will argue that its eigenvalues change in a differentiable manner when one adds a perturbation of the form $\sigma S$, where $S$ is again a self-adjoint linear map on the same space.  
 
This kind of problem is discussed in great length in Kato's classical book "Perturbation Theory for Linear Operators". His methods however require a rather solid understanding of complex function theory.  
 
I will therefore not use Kato's book but the so-called "Feshbach-Schur" method. The idea behind the construction is to introduce a suitable block decomposition of the vector space and then argue that invertibility of $T$ can be reduced to study invertibility of another linear map which is defined on a smaller subspace.  
 
The only requirements of understanding the talk are a good grasp on matrix multiplication, vector and Hilbert spaces and some basic knowledge on the structure of the resolvent set/ the spectrum of a linear map defined on a Hilbert space.  
 
The lecture might be interesting to more advanced students, because the method generalizes to the situation of closed operators $T$ on Hilbert spaces, where in this case several technical assumptions have to be made to give meaning to some definitions and calculations. The basic ideas in the generalized case however remain the same.
Kontaktperson: Thomas Lundsgaard Schmidt