Jordan decomposition of bilinear forms

Let $V$ be an $n$-dimensional vector space over an algebraically closed field $K$ of characteristic 0. Denote by $\sB$ the space of bilinear forms $f:V\times V\to K$. We say that $g\in\sB$ is semisimple if the orbit $\sO_g=\SL_n\cdot g$ is closed in $\sB$, in the Zariski topology. We say that $h\in\sB$ is a null-form if $0\in\overline{\sO_h}$, the Zariski closure of $\sO_h$. 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 $f$.

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Published 1 January 2005