  Singular systems of linear forms were introduced by Khintchine in the 1920s,
and it was shown by Dani in the 1980s that they are in one-to-one
correspondence with certain divergent orbits of one- parameter diagonal groups
on the space of lattices. We give a (conjecturally sharp) upper bound on the
Hausdorff dimension of singular systems of linear forms (equivalently the set
of points with divergent trajectories) as well as the dimension of the set of
points with trajectories 'escaping on average' (a notion weaker than
divergence). This extends work by Cheung, as well as by Chevallier and Cheung,
on the vector case. Our method differs considerably from that of Cheung and
Chevallier, and is based on the method of integral inequalities developed by
Eskin, Margulis and Mozes.
