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18/12/2002, 16:30 — 17:30 — Amphitheatre Pa1, Mathematics Building

Ilya Spitkovsky, *College of William and Mary, Williamsburg, VA*

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Beautiful and mysterious properties of the numerical range, old and
recent

The numerical range (also known as the Hausdorff set) of a matrix
(or an operator on a Hilbert space) is the set of values of the
associated quadratic form on the unit sphere. In this talk, we will
discuss various beautiful properties of the numerical range,
starting with the classical ones (such as its convexity and the
shape in 2-by-2 case) and ending with rather recent: a complete
classification in the case of 3-by-3 matrices, curvature of the
boundary and its influence on the location of the eigenvalues, flat
portions of the boundary (their number and orientation), and an
explanation of why some large (in particular, tridiagonal) matrices
have elliptical numerical ranges.