  We investigate properties of the covariance matrix in the framework of
non-commutative quantum mechanics for an one-parameter family of
transformations between the familiar Heisenberg-Weyl algebra and a particular
extension of it. Employing as a measure of the Robertson-Schr\"{o}dinger
uncertainty principle the linear symplectic capacity of the Weyl ellipsoid (and
its dual), we determine its corresponding bounds. Inequalities between the
capacities for non-commutative phase-spaces are established. We also present a
constructive example based on a simple model to justify our theoretical
predictions.
