## Abstract

We study the distribution of the eigenvalue condition numbers κi=(li∗li)(ri∗ri) associated with real eigenvalues λ_{i} of partially asymmetric N× N random matrices from the real Elliptic Gaussian ensemble. The large values of κ_{i} signal the non-orthogonality of the (bi-orthogonal) set of left l_{i} and right r_{i} eigenvectors and enhanced sensitivity of the associated eigenvalues against perturbations of the matrix entries. We derive the general finite N expression for the joint density function (JDF) P_{N}(z, t) of t=κi2-1 and λ_{i} taking value z, and investigate its several scaling regimes in the limit N→ ∞. When the degree of asymmetry is fixed as N→ ∞, the number of real eigenvalues is O(N), and in the bulk of the real spectrum t_{i}= O(N) , while on approaching the spectral edges the non-orthogonality is weaker: ti=O(N). In both cases the corresponding JDFs, after appropriate rescaling, coincide with those found in the earlier studied case of fully asymmetric (Ginibre) matrices. A different regime of weak asymmetry arises when a finite fraction of N eigenvalues remain real as N→ ∞. In such a regime eigenvectors are weakly non-orthogonal, t= O(1) , and we derive the associated JDF, finding that the characteristic tail P(z, t) ∼ t^{- 2} survives for arbitrary weak asymmetry. As such, it is the most robust feature of the condition number density for real eigenvalues of asymmetric matrices.

Original language | English |
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Pages (from-to) | 309–330 |

Journal | Annales Henri Poincare |

Volume | 2021 |

Issue number | 22 |

DOIs | |

Publication status | Published - 27 Oct 2020 |

## Keywords

- Bi-orthogonal eigenvectors
- Eigenvalue condition numbers
- Weak non-Hermiticity