# Exact Statistical Characterization of $2\times2$ Gram Matrices with Arbitrary Variance Profile

## TL;DR

This paper derives exact statistical characterizations of the $2\times2$ Gram matrix with arbitrary variances, providing new insights into its eigenvalue distribution and implications for dual-antenna communication systems.

## Contribution

It introduces novel exact formulas for the distribution of the Gram matrix and its eigenvalues with arbitrary variance profiles, extending classical Wishart analysis.

## Key findings

- Exact eigenvalue distributions derived for arbitrary variance profiles.
- Simplified expressions facilitate analysis of communication system performance.
- Application to outage data rate analysis in dual-antenna systems.

## Abstract

This paper is concerned with the statistical properties of the Gram matrix $\mathbf{W}=\mathbf{H}\mathbf{H}^\dagger$, where $\mathbf{H}$ is a $2\times2$ complex central Gaussian matrix whose elements have arbitrary variances. With such arbitrary variance profile, this random matrix model fundamentally departs from classical Wishart models and presents a significant challenge as the classical analytical toolbox no longer directly applies. We derive new exact expressions for the distribution of $\mathbf{W}$ and that of its eigenvalues by means of an explicit parameterization of the group of unitary matrices. Our results yield remarkably simple expressions, which are further leveraged to study the outage data rate of a dual-antenna communication system under different variance profiles.