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LARGE SPACES OF MATRICES OF BOUNDED RANK
 

Summary: LARGE SPACES OF MATRICES
OF BOUNDED RANK
By M. D. ATKINSON and S. LLOYD
[Received 2nd November 1979] .
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 classify-
ing 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
form I I where A is a p x q matrix and p + q = r. These examples we

  

Source: Atkinson, Mike - Department of Computer Science, University of Otago

 

Collections: Computer Technologies and Information Sciences