Eigenvector statistics of the product of Ginibre matrices

Area of Science:

• Random Matrix Theory
• Linear Algebra
• Complex Matrix Analysis

Background:

• Understanding the properties of random matrices is crucial in various scientific fields.
• Eigenvector correlations provide insights into the structure of complex matrix products.
• Ginibre and elliptic matrices are fundamental models in random matrix theory.

Purpose of the Study:

• To develop a method for calculating left-right eigenvector correlations.
• To derive analytical expressions for correlation densities.
• To propose and validate a conjecture for integrated vector overlap.

Main Methods:

• Development of a novel analytical method for eigenvector correlation calculation.
• Explicit analytical derivations for small matrix dimensions (N) and product numbers (m).
• Analytical and numerical validation of the proposed conjecture and limiting correlation densities.

Main Results:

• An explicit formula for the integrated overlap between left and right eigenvectors: O=1+(m/2)(N-1).
• Analytical expression for the limiting correlation density for products of Ginibre matrices (N→∞).
• Demonstration that the correlation function for elliptic matrices is independent of their eccentricities.

Conclusions:

• The developed method provides a powerful tool for analyzing eigenvector correlations in matrix products.
• The conjecture for integrated overlap is strongly supported by analytical and numerical evidence.
• The correlation properties of elliptic matrices are robust and independent of specific distribution parameters.