On the eigenvalue distribution of column sub-sampled semi-unitary matrices

Random matrix theory is applied in areas of signal processing, communications, and machine learning. One aspect of random matrix theory involves the study of the eigenvalues of random matrices. For example, in communications, the eigenvalues associated with a channel matrix are used in the analysis of channel capacity and in compressive sensing the eigenvalues of sub-matrices of the sampling matrix can be used to study its restricted isometry property. This paper focuses on a theoretical analysis of the limiting empirical spectral distribution of column sub-sampled semi-unitary matrices. A key contribution of this paper is a closed-form expression for the limiting empirical spectral distribution and the identification of the sufficient conditions for this result.