Invariant Distributions Associated with Matrix Laws Under Structural Symmetry

Let Y(n x m) be a random matrix in the space En x m of real matrices having location-scale parameters (0, Q) such that (i) O belongs to a linear subspace X# of En m; (ii) the distribution Y(Y) exhibits a type of structural symmetry; and (iii) a certain transformation T(Y) is translation-invariant with respect to ? and is invariant under changes of scale. Then the distribution ?{T(Y)} is invariant for all underlying symmetric distributions Y(Y). The types of symmetry are spherical symmetry and symmetry under multiplication on the left by orthogonal and similar matrices; the scale changes are multiplication by a scalar and multiplication on the right by a non-singular matrix. Applications yield the invariance under structural symmetry of distributions usually associated with normal-theory inferences in multivariate regression analysis and in testing equality of dispersion matrices.