Distribution theory of quadratic forms with matrix argument

TL;DR

This paper derives the density and characteristic functions of quadratic forms with matrix arguments for elliptical distributions, including normal, t, and Cauchy, within real normed division algebras.

Contribution

It provides a unified framework for quadratic forms with matrix arguments across various elliptical distributions in real normed division algebras.

Findings

Derived density functions for quadratic forms in elliptical distributions.

Obtained characteristic functions for these quadratic forms.

Unified approach applicable to multiple distribution types.

Abstract

This paper proposes the density and characteristic functions of a general matrix quadratic form $\mathbf{X}^{*}\mathbf{AX}$, when $\mathbf{A} = \mathbf{A}^{*}$, $\mathbf{X}$ has a matrix multivariate elliptical distribution and $\mathbf{X}^{*}$ 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.