Tutorial TUT-7: Finite and infinite random matrix theory

Instructors

A. Edelman; Massachusetts Institute of Technology
R. Rao; Massachusetts Institute of Technology
M. Win; Massachusetts Institute of Technology
M. Chiani; Massachusetts Institute of Technology

Time & Location

Saturday, March 19, 09:00 - 12:00, Location: CC: Room 113-A

Abstract

What are the eigenvalues of my random matrix? This question is becoming increasingly relevant in many signal processing applications. The first step, especially for engineers, while attempting to answer such a question would be to perform numerous Monte-Carlo simulations while gathering empirical evidence about a conjecture or observation. Invariably, the next step would be to ask someone, usually a "specialist in random matrix theory", that might know the answer!

In the latter scenario, the inquirer is often directed to references on finite or infinite random matrix theory. Often, even though the inquirer may have been pointed to the correct set of references, what is known about finite and infinite random matrices is not said in a way that can be easily understood by the non-specialist.

Our motivation for this tutorial is to help demystify random matrix theory and how it has been used in signal processing applications. We want to inform engineers about what mathematicians know or have recently discovered about random matrices; we want to explain, as simply as possible, how some of these theories have been applied; and, we want to help them understand when finite and infinite random matrix theory can be used so that additional applications, which we believe are waiting to be found, can be found.

Tutorial Outline: Our aim is to touch upon various branches of the study of finite and infinite random matrices that are already relevant or are likely to be relevant to the signal processing community. A consequence of this choice is that we will end up lingering on some areas longer than others. Our hope is that this tutorial will confer:

• some familiarity with several of the main thrusts of work in finite and infinite random matrices---sufficient to provide some context for formulating and seeking known solutions to performance analysis questions in signal processing;
• sufficient background and facility to let one read current research publications that apply relatively well-known results from RMT;
• a set of recently developed tools, both analytical and computational, for the analysis of random matrices that may arise in new problems.

Target Audience: This is a tutorial tailored to both graduate students and researchers in academia/industry. Knowledge of linear algebra would be helpful; no prior knowledge of random matrix theory is assumed.

Presenter Information

Alan Edelman is Professor of Mathematics at M.I.T. He has won numerous prizes and awards, including the Householder Prize, the Gordon Bell Prize, and the Chauvenet Prize. His research interests include parallel computing, scientific computing, numerical linear algebra, random eigenvalues, and approximation theory.