  In the present work we show that the joint probability distribution of the
eigenvalues can be expressed in terms of a differential operator acting on the
distribution of some other matrix quantities. Those quantities might be the
diagonal or pseudo-diagonal entries as it is the case for Hermitian matrices.
These representations are called derivative principles. We show them for the
additive spaces of the Hermitian, Hermitian antisymmetric, Hermitian
anti-self-dual, and complex rectangular matrices as well as for the two
multiplicative matrix spaces of the positive definite Hermitian matrices and of
the unitary matrices. In all six cases we prove the uniqueness of the
derivative principles.
