Spaces of Matrices with Several Zero Eigenvalues

Let V be an w-dimensional vector space over some field F, \F\ ^ n, and let SC be a space of linear mappings from V into itself {SC ^ Horn (V, V)) with the property that every mapping has at least r zero eigenvalues. If r = 0 this condition is vacuous but if r = 1 it states that SC is a space of singular mappings; in this case Flanders [1] has shown that d im#"^n(n — 1) and that if dim #" = n(n — 1) then either SC = Hom(K, U) for some (n — l)-dimensional subspace U of V or there exists veV,v ^0 such that SC {X e Horn {V, V): vX = 0}. At the other extreme r = n, the case of nilpotent mappings, a theorem of Gerstenhaber [2] states that dim SC ^ |n(n 1 ) and that if dim SC = $n(n -1) then SC is the full algebra of strictly lower triangular matrices (with respect to some suitable basis of V). In this paper we intend to derive similar results for a general value of r thereby providing some common ground for the above theorems. The condition on the cardinal of F plays an important role in our proofs although we do not know whether it is essential to the theorems. Flanders required also the constraint char F ^ 2 at one point in his proof; we do not require this and in fact our type of argument allows the field in Flanders' theorem to have arbitrary characteristic.