Project Title: Computer Assisted Investigation of
Numerical Ranges
Adviser: Valentin Matache
Description: The numerical range of
an nxn- matrix is the image under the quadratic form
associated to that matrix of the unit sphere in the n-dimensional complex
Euclidean space. A famous theorem of Toeplitz and Hausdorff asserts that a numerical range is always a convex
set. The boundary-curve of a matricial numerical
range can be found based on methods due to Kippenhahn
and Fiedler. In this project we propose that the participating student engages
in the following activities:
A. Write a brief
numerical range primer, based on the following bibliography:
Gustafson and Rao,
Numerical
range : the field of values of linear operators and
matrices, Springer Verlag,
P.R. Halmos, A Hilbert Space Problem Book, Van Nostrand,
This will help her/him understand the research topic and
serve as an introduction to the final, written, research report.
B. Get familiar
with the software called NRANGE.M written by the adviser in MATLAB. The program
graphs numerical ranges of matrices and the participating student is expected
to use it as an experimental tool for discovering properties of numerical
ranges of select classes of matrices and linear operators under the adviser's
guidance.
C. Help the adviser
develop a program that calculates the equation of the boundary curve of a
numerical range in cartesian
coordinates. To that effect, the student
will be expected to study with the adviser's help the following 2 research
papers:
Fiedler Miroslav,
Geometry
of the numerical range of matrices. Linear Algebra Appl. 37 (1981), 81--96.
Keeler, Dennis S.; Rodman,
Leiba; Spitkovsky, Ilya M. The
numerical range of 3x3 matrices. Linear Algebra Appl. 252
(1997), 115--139.
D. Use Halmos's book above to get familiar with Hilbert spaces,
Hilbert-Space operators, and their numerical ranges at a very basic level.
Investigate the shape of those ranges by approximation methods. The adviser
will give more specific guidelines regarding those methods and which classes of
operators are particularly interesting.
E. Put together
her/his findings, written software, significant graphs, and so on in the final
research report to be presented at the
MAM Symposium.
OTHER REQUIREMENTS: The
students interested in the project above are expected to have taken and passed
with maximal grades or close MATH 1950(Calc I), MATH 1960(Calc II), and MATH 4050 (Linear
Algebra ). They should be familiar with computers and MAPLE, be willing to
learn basic MATLAB (a CAS similar to
MAPLE) and TeX, (a popular markup text-editor for
mathematical papers) by taking MATH 3500(Intro to computer assisted
mathematics research) in the spring semester. The student is also expected to
meet with the adviser a couple of times a week, (or communicate actively by
e-mail) for discussions, guidance, and progress reports during the preparation
period of the project.