  The uncertainty relation for continuous variables due to Byalinicki-Birula
and Mycielski expresses the complementarity between two $n$-uples of
canonically conjugate variables $(x_1,x_2,\cdots x_n)$ and $(p_1,p_2,\cdots
p_n)$ in terms of Shannon differential entropy. Here, we consider the
generalization to variables that are not canonically conjugate and derive an
entropic uncertainty relation expressing the balance between any two
$n$-variable Gaussian projective measurements. The bound on entropies is
expressed in terms of the determinant of a matrix of commutators between the
measured variables. This uncertainty relation also captures the complementarity
between any two incompatible linear canonical transforms, the bound being
written in terms of the corresponding symplectic matrices in phase space.
Finally, we extend this uncertainty relation to R\'enyi entropies and also
prove a covariance-based uncertainty relation which generalizes Robertson
relation.
