Skip to main content Accessibility help
×
Hostname: page-component-848d4c4894-tn8tq Total loading time: 0 Render date: 2024-06-15T17:42:28.220Z Has data issue: false hasContentIssue false

2 - Random matrices

from Part I - Theoretical aspects

Published online by Cambridge University Press:  07 October 2011

Romain Couillet
Affiliation:
ST-Ericsson, Sophia Antipolis, France
Mérouane Debbah
Affiliation:
École Supérieure d'Électricité, Gif sur Yvette, France
Get access

Summary

It is often assumed that random matrices is a field of mathematics which treats matrix models as if matrices were of infinite size and which then approximate functionals of realistic finite size models using asymptotic results. We wish first to insist on the fact that random matrices are necessarily of finite size, so we do not depart from conventional linear algebra. We start this chapter by introducing initial considerations and exact results on finite size random matrices.We will see later that, for some matrix models, it is then interesting to study the features of some random matrices with large dimensions. More precisely, we will see that the eigenvalue distribution function FBN of some N × N random Hermitian matrices BN converge in distribution (often almost surely so) to some deterministic limit F when N grows to infinity. The results obtained for F can then be turned into approximative results for FBN, and therefore help to provide approximations of some functionals of BN. Even if it might seem simpler for some to think of F as the eigenvalue distribution of an infinite size matrix, this does not make much sense in mathematical terms, and we will never deal with such objects as infinite size matrices, but only with sequences of finite dimensional matrices of increasing size.

Small dimensional random matrices

We start with a formal definition of a random matrix and introduce some notations.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×