  In this work we apply the density matrices approach to orbital ordering (OO)
in order to study the OO of the spinel vanadates AV$_{2}$O$_{4}$ (A $\equiv$
Zn, Cd and Mg), which is normally believed to be responsible for the structural
transition from cubic to tetragonal phase observed in these compounds. The
density matrices of vanadium atoms are obtained by using {\it state-of-the-art}
full-potential linearized augmented plane wave method based GGA+U calculations.
In the absence of spin-orbit coupling, the present study shows the existence of
anti-ferro OO in the global (local octahedral) coordinate system where $d_{xz}$
and $d_{yz}$ ($d_{xz}$+$d_{yz}$ and $d_{xz}$-$d_{yz}$) orbitals are mainly
occupied at the neighboring V sites for all the compounds.
