On affine spaces of rectangular matrices with constant rank

Let F be a field, and n p r>0 be integers. In a recent article, Rubei has determined, when F is the field of real numbers, the greatest possible dimension for an affine subspace of n--by--p matrices with entries in F in which all the elements have rank r. In this note, we generalize her result to an arbitrary field with more than r+1 elements, and we classify the spaces that reach the maximal dimension as a function of the classification of the affine subspaces of invertible matrices of Mₛ (F) with dimension s2. The latter is known to be connected to the classification of nonisotropic quadratic forms over F up to congruence.