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.

On eigenvector statistics in the spherical and truncated unitary ensembles

“…The matrix of overlaps associated to the bi-orthogonal family of left and right eigenvectors of a non-Hermitian random matrix has been introduced and studied by Chalker & Mehlig in [6,7], then more recently in a series of papers involving a variety of methods from mathematics and physics [1,4,5,8,10,12,14,18,20]. Most of these works deal with Gaussian ensembles, the complex Ginibre ensemble in particular.…”

The Product of $m$ real $N\times N$ Ginibre matrices: Real eigenvalues in the critical regime $m=O(N)$

“…In the critical regime (1.2), the situation becomes somewhat similar to the real elliptic Ginibre ensemble in the weakly asymmetric regime [8], in the sense that a non-trivial portion of the N eigenvalues is real, cf. [6,16].…”

Explicit formulas concerning eigenvectors of weakly non-unitary matrices

“…, which are also the squared eigenvalue condition numbers, are of interest to both the mathematics and physics literature, see for instance [1,2,[8][9][10]13].…”

“…Equation ( 11) together with (12) yields the claim for |v m | 2 and the last column τ m . Plugging these values in (10) gives (13) which is the induction hypotheses with the vector 1 λm v (m−1) . The next lemma, established in a similar way, describes the rows of T under a symmetric assumption.…”