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 \(N\times N\) 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 \(N\rightarrow \infty \). When the degree of asymmetry is fixed as \(N\rightarrow \infty \), 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 \(N\rightarrow \infty \). 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.

我们研究特征值条件数\（\ kappa _i = \ sqrt {（{\ mathbf {l}} _ i ^ * {\ mathbf {l}} _ i）（{\ mathbf {r}} _ i ^ * { \ mathbf {r}} _ i）} \）与来自真实椭圆高斯系综的部分非对称\（N \ N N）个随机矩阵的真实特征值\（\ lambda _i \）相关。\（\ kappa _i \）的较大值表示左侧\（{\ mathbf {l}} _ i \）和右侧\（{\ mathbf {r}} _ i的（双正交）集合的非正交性\）特征向量和相关特征值对矩阵项摄动的增强敏感性。我们导出联合密度函数（JDF）的一般有限N表达式\（{{\ mathcal {P}}} _ N（Z，T）\）的\（T = \卡帕_i ^ 2-1 \）和\（\拉姆达_i \）服用值Ž，并调查其若干缩放限制\（N \ rightarrow \ infty \）中的体制。当不对称度固定为\（N \ rightarrow \ infty \）时，实际特征值的数量为\（\ mathcal {O}（\ sqrt {N}）\），并且在真实频谱的大部分范围内\ （t_i = \ mathcal {O}（N）\），而在接近光谱边缘时，非正交性较弱：\（t_i = \ mathcal {O}（\ sqrt {N}）\）。在这两种情况下，经过适当重新缩放后，相应的JDF与早期研究的完全不对称（Ginibre）矩阵中的JDF一致。当N个特征值的有限分数仍然为\（N \ rightarrow \ infty \）时，会出现不同的弱不对称状态。在这种情况下，特征向量是弱非正交的\（t = \ mathcal {O}（1）\），我们导出了关联的JDF，发现特征尾标\（{{\ mathcal {P}}}（ z，t）\ sim t ^ {-2} \）对于任意弱不对称性仍然存在。这样，对于非对称矩阵的真实特征值，它是条件数密度最强大的功能。