Primitive Spaces of Matrices of Bounded Rank

A weak canonical form is derived for vector spaces of m x n matrices all of rank at most r. This shows that the structure of such spaces is controlled by the structure of an associated 'primitive' space. In the case of primitive spaces it is shown that m and n are bounded by functions of r and that these bounds are tight. 1980 Mathematics subject classification (Amer. Math. Soc.): 15 A 30, 15 A 03. The study of vector spaces % whose vectors are m X n matrices of rank bounded by some number r was begun by Flanders (1962). He showed that such spaces necessarily have dimension at most max(mr, nr) and he classified the spaces of this maximal dimension. His work was extended by Atkinson and Lloyd (1980) to dim % > max(mr, nr) r + 1 while Atkinson and Stephens (1977) treated the case dim % = 2. In this article we shall derive a weaker classification theorem which however is valid for % of arbitrary dimension. This theorem shows that the structure of % depends essentially on an associated 'primitive' space with similar properties to 9C but for which extra information is available. As an application of this result we shall deduce a result which resembles the main theorem of Atkinson and Lloyd (1980) (although it neither implies nor is implied by this theorem). Most of our methods are valid (as in the above works) only when the underlying field has at least r + 1 elements and this condition will be a tacit assumption in all our results. For any space 9C of m X n matrices we let p(%) be the maximum rank of the various matrices in %. If P and Q are non-singular m X m and n X n matrices © Copyright Australian Mathematical Society 1981 The second author gratefully acknowledges the support of the Science Research Council.