Condition Numbers for Real Eigenvalues in the Real Elliptic Gaussian Ensemble

We study the distribution of the eigenvalue condition numbers kappa(i) = root(l(i)(*)l(i))(r(i)*r(i)) associated with real eigenvalues lambda(i) of partially asymmetric N x N random matrices from the real Elliptic Gaussian ensemble. The large values of lambda(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 = kappa(2)(i) - 1 and lambda(i) taking value z, and investigate its several scaling regimes in the limit N -> infinity. When the degree of asymmetry is fixed as N -> infinity, the number of real eigenvalues is O(root N), and in the bulk of the real spectrum t(i) = O( N), while on approaching the spectral edges the non-orthogonality is weaker: t(i) = O(root 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 -> infinity. 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) similar to 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.

Automated summary

The research focuses on the distribution of eigenvalue condition numbers associated with real eigenvalues of partially asymmetric N x N random matrices. It reveals that the asymmetry of the matrices affects the orthogonality of eigenvectors and the sensitivity of eigenvalues against perturbations. The study also shows different scaling regimes and characteristics of the joint density functions in the case of weak and strong asymmetry.