Linear systems of quadratic forms

D. V. Beklemishev

Itogi Nauki i Tekhniki. Seriya Problemy Geometrii. Trudy Geometricheskogo Seminara, Trudy Geometricheskogo Seminara, Tome 3 (1971), pp. 173-192

Résumé

The paper deals with linear systems of quadratic forms. Let $L_n$ and $M_\sigma$ be two complex linear spaces of dimensions $n$ and $\sigma$ respectively. By definition a linear system of quadratic forms is a homomorphism of the symmetric part of $L_n\otimes L_n$ in $M_\sigma$. If we choose a basis in the space $L_n$ and a basis in the space $M_\sigma$, this homomorphism defines $\sigma$ square $n$-matrices $\Lambda_{ij}^\alpha$. ($\alpha=1,\dots,\sigma$; $i,j=1,\dots,n$). We consider the polynom $\det(\rho_\alpha\Lambda_{ij}^\alpha)=0$ and we suppose that it is not identically zero, i.e. the system is not singular. The algebraic variety defined by the equation $\det(\rho_\alpha\Lambda_{ij}^\alpha)=0$ is supposed to have no multiple irreducible component. If these assumptions are true, we can give some necessary and sufficient condition for the existence of a basis in $L_n$ so that all the matrices $\Lambda_{ij}^\alpha$ have the same block-diagonal form. Such a basis is constructed. Some other results are also obtained. In particular it is proved that all the quadratic forms of a singular system of rank $r$ have a common $(n-r)$-dimensional null-space.