Kybernetika

The matrix pencil algebraic characterisation of the families of invariant subspaces of an implicit system $S(F,G): F\dot{z}=Gz\ F, G\in R^{m\times n}$, is further developed by using tools from Exterior Algebra and in particular the Grassmann Representative $g(V)$ of the subspace $V$ of the domain of $(F,G)$. Two different approaches are considered: The first is based on the compound of the pencil $C_d(sF-G)$, which is a polynomial matrix and the second on the compound pencil $sC_d(F)-C_d(G),\ d=\dim V$. For the family of proper spaces of the domain of $(F,G),\ m \geq d$, new characterisations of the invariant spaces $V$ are given in terms of the properties of $g(V)$ as generalised eigenvectors, or invariance conditions for the spaces $\Lambda^p V,\ p=1,2,\ldots,d$.