Condition Numbers for Real Eigenvalues in the Real Elliptic Gaussian Ensemble

Yan V. Fyodorov, Wojciech Tarnowski*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

7 Citations (Scopus)
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We study the distribution of the eigenvalue condition numbers κi=(li∗li)(ri∗ri) associated with real eigenvalues λi of partially asymmetric N× N random matrices from the real Elliptic Gaussian ensemble. The large values of κi signal the non-orthogonality of the (bi-orthogonal) set of left li and right ri 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) PN(z, t) of t=κi2-1 and λi 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 ti= 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 languageEnglish
Pages (from-to)309–330
JournalAnnales Henri Poincare
Issue number22
Publication statusPublished - 27 Oct 2020


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


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