MATH
21 B
Mathematics Math21b Spring 2007
Linear Algebra and Differential Equations
Exhibit:
Course Head: Oliver Knill
Office: SciCtr 434
In your Mathematica lab, you have explored the statistics of the eigenvalues of a large nxn random matrix.
n=1000; A = Table[Random[]-1/2,{n},{n}];
vec[z_]:={Re[z],Im[z]}; EV=Eigenvalues[A];
Show[Graphics[Table[Point[vec[EV[[i]]]],{i,Length[EV]}]],
  AspectRatio->1]
You saw that the eigevanvalues essentially become uniformly distributed on a disc for large n. This is a variant of Girko's Circular law. Also the asymptotic distribution of random symmetric matrices is knonw. Since the eigenvalues are real, this gives a distribution on the line which is called Wigners Semicircle law.


Please send questions and comments to math21b@fas.harvard.edu
Math21b | Oliver Knill | Spring 2007 | Department of Mathematics | Faculty of Art and Sciences | Harvard University