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.
A block-symmetric linearization of odd degree matrix polynomials with optimal eigenvalue condition number and backward error
