TY - JOUR
T1 - Condition Numbers for Real Eigenvalues in the Real Elliptic Gaussian Ensemble
AU - Fyodorov, Yan V.
AU - Tarnowski, Wojciech
PY - 2020/10/27
Y1 - 2020/10/27
N2 - We study the distribution of the eigenvalue condition numbers κi=(li∗li)(ri∗ri) associated with real eigenvalues λi of partially asymmetric N× N random matrices from the real Elliptic Gaussian ensemble. The large values of κi signal the non-orthogonality of the (bi-orthogonal) set of left li and right ri 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) PN(z, t) of t=κi2-1 and λi taking value z, and investigate its several scaling regimes in the limit N→ ∞. When the degree of asymmetry is fixed as N→ ∞, the number of real eigenvalues is O(N), and in the bulk of the real spectrum ti= O(N) , while on approaching the spectral edges the non-orthogonality is weaker: ti=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→ ∞. In such a regime eigenvectors are weakly non-orthogonal, t= O(1) , and we derive the associated JDF, finding that the characteristic tail 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.
AB - We study the distribution of the eigenvalue condition numbers κi=(li∗li)(ri∗ri) associated with real eigenvalues λi of partially asymmetric N× N random matrices from the real Elliptic Gaussian ensemble. The large values of κi signal the non-orthogonality of the (bi-orthogonal) set of left li and right ri 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) PN(z, t) of t=κi2-1 and λi taking value z, and investigate its several scaling regimes in the limit N→ ∞. When the degree of asymmetry is fixed as N→ ∞, the number of real eigenvalues is O(N), and in the bulk of the real spectrum ti= O(N) , while on approaching the spectral edges the non-orthogonality is weaker: ti=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→ ∞. In such a regime eigenvectors are weakly non-orthogonal, t= O(1) , and we derive the associated JDF, finding that the characteristic tail 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.
KW - Bi-orthogonal eigenvectors
KW - Eigenvalue condition numbers
KW - Weak non-Hermiticity
UR - http://www.scopus.com/inward/record.url?scp=85094132772&partnerID=8YFLogxK
U2 - 10.1007/s00023-020-00967-5
DO - 10.1007/s00023-020-00967-5
M3 - Article
AN - SCOPUS:85094132772
SN - 1424-0637
VL - 2021
SP - 309
EP - 330
JO - Annales Henri Poincare
JF - Annales Henri Poincare
IS - 22
ER -