  We present a general framework and procedure to derive uncertainty relations
for observables of quantum systems in a covariant manner. All such relations
are consequences of the positive semidefiniteness of the density matrix of a
general quantum state. Particular emphasis is given to the action of unitary
symmetry operations of the system on the chosen observables, and the covariance
of the uncertainty relations under these operations. The general method is
applied to the case of an $n$-mode system to recover the
  $Sp(2n,\,R)$-covariant multi mode generalization of the single mode
Schr\"{o}dinger-Robertson Uncertainty Principle; and to the set of all
polynomials in canonical variables for a single mode system. In the latter
situation, the case of the fourth order moments is analyzed in detail,
exploiting covariance under the homogeneous Lorentz group $SO(2,\,1)$ of which
the symplectic group $Sp(2,\,R)$ is the double cover.
