| Título: | The Norm of the Product of a Large Matrix and a Random Vector |
| Autores: |
Böttcher, Albrecht; TU Chemnitz Grudsky, Sergei; CINVESTAV del I.P.N. |
| Fecha: | 2003-01-01 |
| Publicador: | Electronic journal of probability |
| Fuente: |
 |
| Tipo: |
Peer-reviewed Article |
| Tema: |
Condition number. Matrix norm. Random vector. Toeplitz matrix. 60H30 (15A60, 47B35, 65F35) |
| Descripción: | 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. |
| Idioma: | No aplica |
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