Distribution theory of quadratic forms for matrix multivariate elliptical distribution

This paper proposes the density and characteristic functions of a general matrix quadratic form X(⁎)AX, when A=A(⁎) is a positive semidefinite matrix, X has a matrix multivariate elliptical distribution and X(⁎) denotes the usual conjugate transpose of 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.