No Eigenvalues Outside the Limiting Support of Generally Correlated Gaussian Matrices

Abla Kammoun, Mohamed-Slim Alouini

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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. Following the approach proposed in [1], we prove that under some mild conditions, there is no eigenvalue outside the limiting support of generally correlated Gaussian matrices. As an outcome of this result, we establish that the smallest singular value of these matrices is almost surely greater than zero. From a practical perspective, this control of the smallest singular value is paramount to applications from statistical signal processing and wireless communication, in which this kind of matrices naturally arise.
Original languageEnglish (US)
Pages (from-to)4312-4326
Number of pages15
JournalIEEE Transactions on Information Theory
Issue number7
StatePublished - May 4 2016

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