Distribution theory of quadratic forms with matrix argument

Abstract

This paper proposes the density and characteristic functions of a general matrix quadratic form ${\mathbf{X}}^{\ast }\mathbf{AX}$, when $\mathbf{A}={\mathbf{A}}^{\ast }$, $\mathbf{X}$ has a matrix multivariate elliptical distribution and ${\mathbf{X}}^{\ast }$ denotes the usual conjugate transpose of $\mathbf{X}$. These results are obtained for real normed division algebras. With particular cases we obtained the density and characteristic functions of matrix quadratic forms for matrix multivariate normal, Pearson type VII, $t$ and Cauchy distributions.

17 pages. arXiv admin note: text overlap with arXiv:1207.1993, arXiv:1010.1799, arXiv:1012.5856, arXiv:1011.5083