A note on some extensions of the matrix angular central Gaussian distribution

TL;DR

This paper extends the matrix angular central Gaussian distribution to the complex domain, analyzing the distribution of orientations of complex random matrices and their transformations.

Contribution

It introduces the complex MACG distribution, deriving its properties and exploring related distribution families for complex matrices.

Findings

Complex MACG distribution is derived from complex Gaussian matrices.

Orientation of transformed complex matrices also follows CMACG.

The paper characterizes distribution families leading to CMACG.

Abstract

This paper extends the notion of the matrix angular central distribution (MACG) to the complex case. We start by considering the normally distributed random complex matrix ($Z$) and show that is the orientation ($H_Z=Z(Z'Z)^{-1}$) has complex MACG (CMACG) distribution. Then we discuss the distribution of the orientation of the linear transformation of the random matrix which orientation part has CMACG distribution. Finally, we discuss the family of distributions which lead to the CMACG distribution.