On uncertainty relations for vector-valued operators

Just as the Heisenberg uncertainty principle for one-dimensional quantum-mechanical quantities can be expressed in terms of a variance by means of Weyl's inequality, for multidimensional quantities this principle can be expressed in terms of generalized variances and covariation matrices by means of inequalities analogous to Weyl's. These inequalities are proved abstractly in this paper: not only for physical vector quantities but also for arbitrary vector-valued operators with commutting self-adjoint components.