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

   
@article{EJP132,
	author = {Albrecht Böttcher and Sergei Grudsky},
	title = {The Norm of the Product of a Large Matrix and a Random Vector},
	journal = {Electron. J. Probab.},
	fjournal = {Electronic Journal of Probability},
	volume = {8},
	year = {2003},
	keywords = {Condition number. Matrix norm. Random vector. Toeplitz matrix.},
	abstract = {Given a real or complex $n \times n$ matrix $A_n$, we compute the expected value and the variance of the random variable $\| A_n x\|^2/\| A_n \|^2$, where $x$ is uniformly distributed on the unit sphere of $R^n$ or $C^n$. The result is applied to several classes of structured matrices. It is in particular shown that if $A_n$ is a Toeplitz matrix $T_n(b)$, then for large $n$ the values of $\| A_n x\|/\| A_n \|$ cluster fairly sharply around $\| b \|_2/\| b \|_\infty$ if $b$ is bounded and around zero in case $b$ is unbounded.},
	pages = {no. 7, 1-29},
	issn = {1083-6489},
	doi = {10.1214/EJP.v8-132},    
        url = {http://ejp.ejpecp.org/article/view/132}}