LARGE SPACES OF MATRICES OF BOUNDED RANK

IN THIS paper we consider subspaces X of M^*, the space of all m x n matrices with entries in some given field, with the property that each matrix of X has rank at most r. In [2] Flanders showed that such spaces necessarily have dimension at most max (mr, nr) and he determined the spaces of precisely this dimension. We shall extend this work by classifying the spaces of dimension slightly lower than this upper bound. Our results depend on the (often unstated) assumption that the ground field has at least r +1 elements but, unlike Flanders, we do not need to exclude the characteristic 2 case. If every matrix in the space X has rank at most r the same is clearly true of the space PXQ = {PXQ: X e X} where P, Q are non-singular mxtn, nxn matrices respectively. This equivalent space PXQ can also be derived from X by performing row and column operations to all matrices of X simultaneously. A wide class of examples is provided by spaces equivalent to subspaces of the space 9i(p, q) of all matrices of the