On the Eigenvalue Density of Real and Complex Wishart Correlation Matrices

TL;DR

This paper derives an exact formula for the eigenvalue density of real Wishart correlation matrices using supersymmetry, overcoming previous mathematical obstacles and extending the known results from complex to real cases.

Contribution

It introduces a supersymmetry-based method to obtain the eigenvalue density for real Wishart matrices, which was previously intractable.

Findings

Exact formula for real eigenvalue density derived

Formula expressed as twofold integrals and finite sums

Advances understanding of real correlation matrices

Abstract

Wishart correlation matrices are the standard model for the statistical analysis of time series. The ensemble averaged eigenvalue density is of considerable practical and theoretical interest. For complex time series and correlation matrices, the eigenvalue density is known exactly. In the real case, however, a fundamental mathematical obstacle made it forbidingly complicated to obtain exact results. We use the supersymmetry method to fully circumvent this problem. We present an exact formula for the eigenvalue density in the real case in terms of twofold integrals and finite sums.