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.