Statistical properties of eigenvectors in non-Hermitian Gaussian random matrix ensembles

Statistical properties of eigenvectors in non-Hermitian random matrix ensembles are discussed, with an emphasis on correlations between left and right eigenvectors. Two approaches are described. One is an exact calculation for Ginibre’s ensemble, in which each matrix element is an independent, identically distributed Gaussian complex random variable. The other is a simpler calculation using N−1 as an expansion parameter, where N is the rank of the random matrix: this is applied to Girko’s ensemble. Consequences of eigenvector correlations which may be of physical importance in applications are also discussed. It is shown that eigenvalues are much more sensitive to perturbations than in the corresponding Hermitian random matrix ensembles. It is also shown that, in problems with time evolution governed by a non-Hermitian random matrix, transients are controlled by eigenvector correlations.

[1]  The U(1) Problem in Chiral Random Matrix Models , 1996, hep-lat/9611012.

[2]  Correlations of eigenvectors for non-Hermitian random-matrix models. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[3]  David R. Nelson,et al.  NON-HERMITIAN LOCALIZATION AND POPULATION BIOLOGY , 1997, cond-mat/9708071.

[4]  K. Efetov.  Directed Quantum Chaos , 1997, cond-mat/9702091.

[5]  Lloyd N. Trefethen,et al.  Pseudospectra of Linear Operators , 1997, SIAM Rev..

[6]  H. Sommers,et al.  Statistics of complex levels of random matrices for decaying systems , 1992 .

[7]  Sommers,et al.  Spectrum of large random asymmetric matrices. , 1988, Physical review letters.

[8]  Anne E. Trefethen,et al.  Hydrodynamic Stability Without Eigenvalues , 1993, Science.

[9]  Spectrum of the fokker-planck operator representing diffusion in a random velocity field , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[10]  Nelson,et al.  Localization Transitions in Non-Hermitian Quantum Mechanics. , 1996, Physical review letters.

[11]  M. Wilkinson.  Statistics of multiple avoided crossings , 1989 .

[12]  I. Goldsheid,et al.  DISTRIBUTION OF EIGENVALUES IN NON-HERMITIAN ANDERSON MODELS , 1997, cond-mat/9707230.

[13]  Maciej A. Nowak,et al.  Non-Hermitian random matrix models: Free random variable approach , 1997 .

[14]  Y. Fyodorov,et al.  ALMOST HERMITIAN RANDOM MATRICES : CROSSOVER FROM WIGNER-DYSON TO GINIBRE EIGENVALUE STATISTICS , 1997, cond-mat/9703152.

[15]  Y. Fyodorov,et al.  STATISTICAL PROPERTIES OF EIGENFUNCTIONS OF RANDOM QUASI 1D ONE-PARTICLE HAMILTONIANS , 1994 .

[16]  B. Mehlig,et al.  EIGENVECTOR STATISTICS IN NON-HERMITIAN RANDOM MATRIX ENSEMBLES , 1998 .

[17]  A. Zee,et al.  Non-hermitian random matrix theory: Method of hermitian reduction , 1997 .

[18]  J. Ginibre.  Statistical Ensembles of Complex, Quaternion, and Real Matrices , 1965 .

[19]  Sommers,et al.  Eigenvalue statistics of random real matrices. , 1991, Physical review letters.