Eigenvalue distributions for some correlated complex sample covariance matrices

TL;DR

This paper derives determinant formulas for the eigenvalue distributions of correlated complex sample covariance matrices, improving computational efficiency over previous methods that used sums over partitions and Schur polynomials.

Contribution

It provides new determinant-based expressions for eigenvalue distributions of correlated complex sample covariance matrices, enhancing computational efficiency.

Findings

Eigenvalue distributions expressed as m x m determinants

More efficient computation compared to previous sum-based methods

Applicable to matrices with row and column correlations

Abstract

The distributions of the smallest and largest eigenvalues for the matrix product $Z^\dagger Z$, where $Z$ is an $n \times m$ complex Gaussian matrix with correlations both along rows and down columns, are expressed as $m \times m$ determinants. In the case of correlation along rows, these expressions are computationally more efficient than those involving sums over partitions and Schur polynomials reported recently for the same distributions.