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The Newton Polygon and Eigenvalue Perturbation Theory.Colloquium
|Speaker: ||Prof. Jim Burke, Mathematics, University of Washington|
|Location: ||693 Kerr|
|Start time: ||Mon, Apr 10 2000, 4:10PM|
Newton was a master of computation. Indeed, many modern
computational techniques find their roots in
Newton's work. In the first half of this talk,
I will recount Newton's beautifully
simple trick for computing the roots of a perturbed polynomial
as a function of the perturbation.
The trick is called the Newton polygon. This part
of the talk is accessible to students having a working
knowledge of College Algebra. After introducing the method, I
will use Newton's notes to give some indication of why it works.
The second half of the talk will discuss how Newton's polygon
can be used to derive the Lidskii-Vishik-Lyusternik perturbation
theory for the eigenvalues of matrices with arbitrary Jordan
structure. I then show how this approach through
the Newton polygon can be used to extend Lidskii's results to some
non-generic cases where the standard theory does not apply.
This half of the talk is accessible to senior level undergraduates
and graduate students having a working knowledge of the Jordan form
for matrices over the complex numbers.
This talk is based on a collaboration with Julio Moro of the
Universidad Carlos III, Madrid, and Michael Overton of the