Spaces of Matrices of Bounded Rank

IN this paper we shall consider matrices over a field F and shall prove the following result: THEOREM. Let M be a 2-dimensional space o/mxn matrices with the property that rank (X) *£ k < \F\ for every XeM. Then there exist two integers r, s, O^r, s *£ fc with r + s = k, and two non-singular matrices P, Q such that, for all XeM, PXQ has the form Notice that a matrix of the above form necessarily has rank at most k and so, apart from the restriction \F\ > fc, our theorem essentially charac-terises such 2-dimensional subspaces. We may interpret the matrices as the matrices of linear transformations or of bilinear forms and this gives the following two equivalent forms of the theorem valid for finite dimensional vector spaces: Our interest in this result stems from its application to computational complexity [1] but conditions like those of the theorem have been studied before [2,3]. Indeed in [3] the conclusion of our theorem is proved under the stronger assumptions rank (X) = k for all Xe M and F is algebraically closed. However our result, besides being more general, has a shorter proof. We note that, in general, the conditions in the theorem cannot be weakened. The condition \F\ > k is necessary on account of the space "1 1 0_ and 0 1 1 generated by 1 and 1 over GF(2) and the condition