Jordan decomposition of bilinear forms

摘要

Let V be an n-dimensional vector space over an algebraically closed field K of characteristic 0. Denote by 5 the space of bilinear forms f : V x V -> K. We say that g is an element of beta is semisimple if the orbit O-g = SLn center dot g is closed in beta, in the Zariski topology. We say that h is an element of beta is a null-form if 0 is an element of U (O-h) over bar, the Zariski closure of O-h. 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.