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Dominik Schröder

On the condition number of the shifted real Ginibre ensemble

Giorgio Cipolloni, László Erdős, Dominik Schröder

SIAM J. Matrix Anal. Appl.Vol. 43 (2022)


We show that complex shifts of real matrices have a stronger regularizing effect than real shifts. As a consequence we obtain improved bounds on the practically relevant condition number and demonstrate that our results give sharp asymptotics for the running time of the CG algorithm.


We derive an accurate lower tail estimate on the lowest singular value σ1(Xz)\sigma_1(X-z) of a real Gaussian (Ginibre) random matrix XX shifted by a complex parameter zz. Such shift effectively changes the upper tail behaviour of the condition number κ(Xz)\kappa(X-z) from the slower P(κ(Xz)t)1/t\mathbf{P}(\kappa(X-z)\ge t)\lesssim 1/t decay typical for real Ginibre matrices to the faster 1/t21/t^2 decay seen for complex Ginibre matrices as long as zz is away from the real axis. This sharpens and resolves a recent conjecture in [arXiv:2005.08930] on the regularizing effect of the real Ginibre ensemble with a genuinely complex shift. As a consequence we obtain an improved upper bound on the eigenvalue condition numbers (known also as the eigenvector overlaps) for real Ginibre matrices. The main technical tool is a rigorous supersymmetric analysis from our earlier work [arXiv:1908.01653].