An Invariant Property of the Elliptically Contoured Distribution about the Non-Singular Matrix Transformation

In this paper, we first extend the definitions of matrix F and t distributions to the left spherical distribution family, prove the density functions have no relation with the one producing them and then show that discuss the elliptically contoured distributions are invariant under nonsingular matrix transformations. These distributions include the matrix Beta, inverse Beta, Dirichlet, inverse Dirichlet, F and t etc. And finally it is shown that their distribution density functions not only have no relation with the density function generating them but also the transformation matrix.