Condition Numbers for Real Eigenvalues in the Real Elliptic Gaussian Ensemble

Abstract

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