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 Electronic Journal of Probability > Vol. 8 (2003) > Paper 7 open journal systems 


The Norm of the Product of a Large Matrix and a Random Vector

Albrecht Böttcher, TU Chemnitz
Sergei Grudsky, CINVESTAV del I.P.N.


Abstract
Given a real or complex n x n matrix A, let X be the random variable ||Ax||^2 divided by ||A||^2 where x is uniformly distributed on the unit sphere of Rn or Cn.
We compute the expected  value and the variance of the random variable X. The result is applied to several classes of structured matrices. It is in particular shown that
if  A is a Toeplitz matrix, then for large n the values of X cluster fairly sharply around a number that can be completely identified.


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Pages: 1-29

Published on: May 22, 2003
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Electronic Journal of Probability. ISSN: 1083-6489