Squared eigenvalue condition numbers and eigenvector correlations from the single ring theorem
Serban Belinschi, Maciej A Nowak, Roland Speicher, Wojciech Tarnowski · Journal of Physics A Mathematical and Theoretical · 2016
Abstract We extend the so-called ‘single ring theorem’ (Feinberg and Zee 1997 Nucl. Phys . B 504 579), also known as the Haagerup–Larsen theorem (Haagerup and Larsen 2000 J. Funct. Anal . 176 331). We do this by showing that in the limit when the size of the matrix goes to infinity a particular correlator between left and right eigenvectors of the relevant non-hermitian matrix X , being the spectral density weighted by the squared eigenvalue condition number, is given by a simple formula involving only the radial spectral cumulative distribution function of X . We show that this object allows the calculation of the conditional expectation of the squared eigenvalue condition number. We give examples and provide a cross-check of the analytic prediction by the large scale numerics.