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


 
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1. Title Title of document The Norm of the Product of a Large Matrix and a Random Vector
 
2. Creator Author's name, affiliation, country Albrecht Böttcher; TU Chemnitz
 
2. Creator Author's name, affiliation, country Sergei Grudsky; CINVESTAV del I.P.N.
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Condition number. Matrix norm. Random vector. Toeplitz matrix.
 
3. Subject Subject classification 60H30 (15A60, 47B35, 65F35)
 
4. Description 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.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s)
 
7. Date (YYYY-MM-DD) 2003-05-22
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/132
 
10. Identifier Digital Object Identifier 10.1214/EJP.v8-132
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 8
 
12. Language English=en
 
14. Coverage Geo-spatial location, chronological period, research sample (gender, age, etc.)
 
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