Condition Numbers for Real Eigenvalues in the Real Elliptic Gaussian Ensemble

Abstract

We study the distribution of the eigenvalue condition numbers kappa _i=sqrt{ ({mathbf{l}}_i^* {mathbf{l}}_i)({mathbf{r}}_i^* {mathbf{r}}_i)} associated with real eigenvalues lambda _i of partially asymmetric Ntimes N random matrices from the real Elliptic Gaussian ensemble. The large values of kappa _i signal the non-orthogonality of the (bi-orthogonal) set of left {mathbf{l}}_i and right {mathbf{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) {{mathcal {P}}}_N(z,t) of t=kappa _i^2-1 and lambda _i taking value z, and investigate its several scaling regimes in the limit Nrightarrow infty . When the degree of asymmetry is fixed as Nrightarrow infty , the number of real eigenvalues is mathcal {O}(sqrt{N}), and in the bulk of the real spectrum t_i=mathcal {O}(N), while on approaching the spectral edges the non-orthogonality is weaker: t_i=mathcal {O}(sqrt{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 Nrightarrow infty . In such a regime eigenvectors are weakly non-orthogonal, t=mathcal {O}(1), and we derive the associated JDF, finding that the characteristic tail {{mathcal {P}}}(z,t)sim 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.