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Numerical range for random matrices
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Numerical range for random matrices
Bibliographic description
| title: | Numerical range for random matrices |
| author: | Collins Benoît, Gawron Piotr, Litvak Alexander E., Życzkowski Karol 
|
| Journal title: | Journal of Mathematical Analysis and Applications |
| volume: | 418 |
| issue: | 1 |
| date of publication: | 2014 |
| pages: | 516-533 |
| ISSN: |
0022-247X |
| eISSN: |
1096-0813 |
| DOI: | |
| language: | English |
| Journal language: | English |
| abstract in english: | We analyze the numerical range of high-dimensional random matrices, obtaining limit results and corresponding quantitative estimates in the non-limit case. For a large class of random matrices their numerical range is shown to converge to a disc. In particular, numerical range of complex Ginibre matrix almost surely converges to the disk of radius √2. Since the spectrum of non-hermitian random matrices from the Ginibre ensemble lives asymptotically in a neighborhood of the unit disk, it follows that the outer belt of width √2 - 1 containing no eigenvalues can be seen as a quantification the non-normality of the complex Ginibre random matrix. We also show that the numerical range of upper triangular Gaussian matrices converges to the same disk of radius √2, while all eigenvalues are equal to zero and we prove that the operator norm of such matrices converges to √2e. |
| keywords in english: | Ginibre ensemble, Gaussian random matrix, GUE, numerical range, field of values, triangular random matrix |
| number of pulisher's sheets: | 1 |
| affiliation: | Wydział Fizyki, Astronomii i Informatyki Stosowanej : Instytut Fizyki im. Mariana Smoluchowskiego |
| type: | journal article |
| subtype: | academic paper |
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