The Density Functions of the Singular Quaternion Normal Matrix and the Singular Quaternion Wishart Matrix

Abstract

In this article, we give the density functions of the singular quaternion normal matrix and the singular quaternion Wishart matrix. Furthermore, we also give the density functions of certain singular quaternion -matrix and the singular quaternion F -matrix in terms of the density function of the singular quaternion Wishart matrix and hypergeometric functions of quaternion matrix argument.

Jacobians of singular matrix transformations: Extensions

“…i.e., they study densities and then calculate the Jacobians of the matrix transformations with respect to the Hausdorff measure, see Khatri (1968), Uhlig (1994), , , Srivastava (2003), , Díaz-García and González-Farías (2005a), Díaz-García and González-Farías (2005b), Ip et al (2007) and Díaz-García (2007), among others. In addition, studied some of these latter Jacobians in the singular complex case and Li and Xue (2010) considered the singular quaternion case.…”

On the real representation of quaternion random variables

“…They used a Fourier transform on the results given by Andersson [1] to yield explicit expressions for the probability density and characteristic functions of the quaternion Wishart distribution. The non-central quaternion Wishart distribution was discussed by Kabe [12], while Li and Xue [14] derived the singular quaternion Wishart distribution. More technical results, specifically regarding Selberg-type squared matrices, gamma and beta integrals are found in the paper by Gupta and Kabe [9].…”

“…Noting the fact that Σ and R are Hermitian, that the conjugate of Re tr is again Re tr,and that according to (1) the arguments of Re tr may be rearranged, (14) reduces to…”

“…The vec operator is frequently used in expressions involving matrices of quaternions, see for instance Li and Xue [14], and is defined as vecZ…”

“…The results in this paper should therefore not be seen as corollaries of the work in real normed division algebras (see for example [5], [6], [7], [13] and [14]), but the significant contribution that is made in the statistical context, by using the real counterpart approach.…”