Chennai Mathematical Institute


3.30 pm, Seminar Hall
CMI Silver Jubilee Lecture
How likely is a random matrix to be singular?

Manjunath Krishnapur
IISc, Bangalore.


Let A be a random nxn matrix chosen uniformly from the set of all 2^{n^2} matrices whose entries are 0 or 1. Let p_n be the probability that A is singular. It is an open question to find the asymptotics of p_n, although it is believed to asymptotically behave as 2^{-n}. In this lecture I mention this problem, and give a proof due to Komlos that p_n-->0. This leads us to a very interesting type of problem in probability/additive combinatorics called the Littlewood-Offord problem, and I shall talk about that too. This is an expository lecture for undergraduates. It should be accessible to anyone who knows/can learn what independent Bernoulli random variables are.