  We extend Hardy's uncertainty principle for a square integrable function and
its Fourier transform to the multidimensional case using a symplectic
diagonalization. We use this extension to show that Hardy's uncertainty
principle is equivalent to a statement on the Wigner distribution of the
function. We give a geometric interpretation of our results in terms of the
notion of symplectic capacity of an ellipsoid. Furthermore, we show that
Hardy's uncertainty principle is valid for a general Lagrangian frame of the
phase space. Finally, we discuss an extension of Hardy's theorem for the Wigner
distribution for exponentials with convex exponents.
