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Mathematics Department Técnico Técnico

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18/12/2002, 16:30 — 17:30 — Amphitheatre Pa1, Mathematics Building
, College of William and Mary, Williamsburg, VA

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.

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The Mathematics Colloquium is a series of monthly talks organized by the Department of Mathematics of IST, aiming to be a forum for the presentation of mathematical ideas or ideas about Mathematics. The Colloquium welcomes the participation of faculty, researchers and undergraduate or graduate students, of IST or other institutions, and is seen as an opportunity of bringing together and fostering the building up of ideas in an informal atmosphere.

Organizers: Conceição Amado, Lina Oliveira e Maria João Borges.