Distribution theory of quadratic forms with matrix argument

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