  Design matrices are sparse matrices in which the supports of different
columns intersect in a few positions. Such matrices come up naturally when
studying problems involving point sets with many collinear triples. In this
work we consider design matrices with block (or matrix) entries. Our main
result is a lower bound on the rank of such matrices, extending the bounds
proved in {BDWY12,DSW12} for the scalar case. As a result we obtain several
applications in combinatorial geometry. The first application involves
extending the notion of structural rigidity (or graph rigidity) to the setting
where we wish to bound the number of `degrees of freedom' in perturbing a set
of points under collinearity constraints (keeping some family of triples
collinear). Other applications are an asymptotically tight Sylvester-Gallai
type result for arrangements of subspaces (improving {DH16}) and a new
incidence bound for high dimensional line/curve arrangements.
  The main technical tool in the proof of the rank bound is an extension of the
technique of matrix scaling to the setting of block matrices. We generalize the
definition of doubly stochastic matrices to matrices with block entries and
derive sufficient conditions for a doubly stochastic scaling to exist.
