Linear Spaces of Toeplitz and Nilpotent Matrices

Let F be an arbitrary field and let J¢/', (F) denote the linear space of all matrices of order n over F. Let W be a subspace of J/t, (F). Let r be an integer with 1 ~< r ~ n. Flanders [21 considered the question of how large the dimension of W must be to guarantee that W contains a matrix whose rank is at least equal to r. There exist spaces W of dimension ( r 1 ) n containing no matrix of rank r or greater. Flanders showed that if F has at least r elements and dim(W)~> ( r 1)n + 1, then W contains a matrix A with rank(A)~>r. Meshu lam[9] gave a simpler proof of Flander's theorem, which did not require any restriction on the size of the field F. Meshulam's proof of Flanders' theorem applies more generally to the following situation. Let C = [co ] be a (0, 1)-matrix of order n and let d/t, [C] (F) denote the coordinate subspace of ~n (F) of matrices J (= [xo ] such that c / j=0 implies xi j=0. Let d,(C) be the largest number of l's of C which are contained in the union of r lines ~ of C. Theorem 1 of [9] implies the following result.