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

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

1999-06-17

arxiv.org/abs/cond-mat/9906279v1

J. Math. Phys. 41, 3233 (2000)

Physics

Condensed Matter

Disordered Systems and Neural Networks

Scientific paper

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.

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