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Mathematics Math21b Fall 2010

Linear Algebra and Differential Equations

Exhibit: Snow flake spectra

Course Head: Oliver Knill

Office: SciCtr 434

Email: knill@math.harvard.edu

Movie by Oliver Knill October 10, 2010. This is a bit improved on a movie shown first during a talk given at the Harvard Dunster House on September 15, 2010. Tap here with an ipod/iphone/ipad.

These are examples of spectra of random matrices in which one parameter is changed.
The random variables are not independent however. There are strong correlations between
the matrix entries. The situation with independent random variables is
more boring. Each point is an eigenvalue. When the parameter changes in the probability space, the eigenvalues move like a swarm of 2000 particles, fighting each other under a changing external force. Eigenvalues "fear" each other. Since each eigenvalue is a hero "battling" in the complex plane, a heroic music theme has been chosen. More about this in the Mathematica project. Added October 17: the spectra seen here are expected to have a fractal structure if the matrix size goes to infinity. It was Mandelbrot, who just died who had popularized the subject of fractals. Mandelbrot had lived at 75 Cambridge street in Cambridge (pretty close to the Science museum). He had been a visiting professor at the Harvard mathematics department. Added October 31: here is a picture of the spectrum of a 30'000 x 30'000 matrix computed on Odyssee. |

Please send questions and comments to math21b@fas.harvard.edu

Math21b (Exam Group 1)| Oliver Knill | Fall 2010 |
Department of Mathematics |
Faculty of Art and Sciences |
Harvard University