On the Smallest Eigenvalue of General correlated Gaussian Matrices

TL;DR

This paper studies the spectral properties of correlated Gaussian matrices with independent columns, showing that under certain conditions, their smallest singular value remains bounded away from zero as matrix dimensions grow large.

Contribution

It provides a rigorous proof that the smallest eigenvalue of such matrices does not approach zero under specific correlation structures and growth regimes.

Findings

Smallest eigenvalue remains positive asymptotically

Conditions for eigenvalue boundedness are established

Applicable to signal processing and wireless communication models

Abstract

This paper investigates the behaviour of the spectrum of generally correlated Gaussian random matrices whose columns are zero-mean independent vectors but have different correlations, under the specific regime where the number of their columns and that of their rows grow at infinity with the same pace. This work is, in particular, motivated by applications from statistical signal processing and wireless communications, where this kind of matrices naturally arise. Following the approach proposed in [1], we prove that under some specific conditions, the smallest singular value of generally correlated Gaussian matrices is almost surely away from zero.