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.
Computing the Fréchet Derivative of the Matrix Logarithm and Estimating the Condition Number