  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.
