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Mathematics Math21b Spring 2007

Linear Algebra and Differential Equations

Exhibit:

Course Head: Oliver Knill

Office: SciCtr 434

Email: knill@math.harvard.edu

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