On the Condition Number of a Randomly Perturbed Matrix

COMPUTER SCIENCE/DISCRETE MATH I
Topic:On the Condition Number of a Randomly Perturbed Matrix
Speaker:Van Vu
Affiliation:Rutgers, The State University of New Jersey
Date:Monday, January 22
Time/Room:11:15am - 12:15pm/West Building Lecture Theatre

Let $M$ be an arbitrary $n$ by $n$ matrix. We study the condition number a random perturbation $M+N_n$ of $M$, where $N_n$ is a random matrix, motivated by a problem raised by Spielman and Teng. It is shown that, under very general conditions on $M$ and $N_n$, the condition number of $M+N_n$ is polynomial in $n$ with very high probability. The main novelty here is that we allow $N_n$ to have discrete distributions. Discreteness posed a considerable mathematical challenge which we were able to overcome by bringing in ideas and tools from additive combinatorics. The core of our proof is the so-called Inverse Littlewood-Offord theorem which characterizes all integer sequences with many colliding subsums. Joint work with T. Tao (UCLA)