Jordan decomposition of bilinear forms

Abstract

Let V be an n-dimensional vector space over an algebraically closed field K of characteristic 0. Denote by B the space of bilinear forms f : V ×V → K. We say that g ∈ B is semisimple if the orbit Og = SLn · g is closed in B, in the Zariski topology. We say that h ∈ B is a null-form if 0 ∈ Oh, the Zariski closure of Oh. We introduce the Jordan decomposition for bilinear forms f = g + h (g semisimple, h a null-form) in analogy with the well known Jordan decomposition for linear operators. While the latter decomposition is unique, this is not the case for the former. If f is not a null-form, we introduce the primary decomposition of f and use it to construct all possible Jordan decompositions of f .