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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