- latex,
- postscript, and

In this lecture, we bounded the largest singular value of a random matrix, using a 1-net. We also saw a demo of the instability of a bad example for Gaussian Elimination. The idea for the proof presented in class came from:

- "Spaces with Large Distance to l^n_inf and Random Matrices", by Stanislaw J. Szarek, American Journal of Mathematics, Vol 112, Issue 6, Dec. 1990, pp. 899-942. (available at JSTOR).

- "A Limit Theorem for the Norm of Random Matrices", bu Stuart Geman, appeared in Annals of Probability, Vol 8, Issue 2 (Apr. 1980), pp. 252-261. (available at JSTOR).
- "Condition numbers of random matrices", by Stanislaw J. Szarek, appared in the Journal of Complexity, 7(2):131-149, June 1991.
- "Eigenvalues and Condition Numbers of Random Matrices", by Alan Edelman, appeared in SIAM J. Matrix Anal. Appl., 1988, vol 9, no 4, pp. 543-560.

In class, Igor Pak mentioned a paper that bounds the probability that a random +/-1 matrix is degenerate. The paper is:

- "On the probability that a random +/- 1 matrix is singular", by Jeff Kahn, Janos Komlos, and Endre Szemeredi. It appeared in the Journal of the American Mathematical Society, Vol 8, Issue 1 (Jan 1995), pp. 223-240. (available at JSTOR)

All of the matlab code used in this lecture may be found on the BAP matlab code directory. Today's matlab example was:

A = kahan2(100); b = ones(100,1); x = A \ b; format short g A*x norm(A*x - b) Ap = A + randn(100)/(10^7); x = Ap \ b; A*x norm(A*x - b)