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Entanglement of eigenvalues and eigenvectors in free non-hermitian matrices

Banff International Research Station for Mathematical Innovation and Discovery

We start from recalling the generalization of the R-transform for strictly-nonhermitian large random matrices. Then we present the generalization of the Voiculescu equation for nonhermitian operators, pointing at the entangled coevolution of the spectra and eigenvectors. We exemplify this evolution solving the simplest, Gaussian case (Ginibre ensemble). Finally, basing on the recent work by S. Belinschi, M.A. Nowak, R. Speicher and W. Tarnowski, we show how to augment the single ring theorem (Haagerup-Larsen theorem) with the predictions for certain left-right eigenvector correlations. We conclude with presentation of few examples (products/ratios of Ginibre ensembles, arbitrary product of truncated unitary matrices , sum of arbitrary number of unitary ensembles).

Mathematics; Functional analysis; Probability theory and stochastic processes; Operator theory/algebras

video/mp4

Author affiliation: Jagiellonian University

2017-05-15

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/61646

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/