Abstract of a paper by Benoit Collins, Piotr Gawron, Alexander E. Litvak, and Karol Zyczkowski
This is an announcement for the paper "Numerical range for random matrices" by Benoit Collins, Piotr Gawron, Alexander E. Litvak, and Karol Zyczkowski. Abstract: We analyze the numerical range of high-dimensional random matrices, obtaining limit results and corresponding quantitative estimates in the non-limit case. We show that the numerical range of complex Ginibre ensemble converges to the disk of radius $\sqrt{2}$. 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 $\sqrt{2}-1$ 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 $\sqrt{2}$, while all eigenvalues are equal to zero and we prove that the operator norm of such matrices converges to $\sqrt{2e}$. Archive classification: math.OA math.FA math.PR quant-ph Mathematics Subject Classification: 5A60, 47A12, 15B52 (primary), 46B06, 60B20 (secondary) Remarks: 22 pages, 4 figures Submitted from: gawron@iitis.pl The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1309.6203 or http://arXiv.org/abs/1309.6203
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