## International Federation of Automatic Control |

## Session WE1: Algebraic MethodsChair: Nicos Karcanias - United Kingdom Co-chair: Petr Husek - Czech Republic Room: Assunção 2007-10-17 From 16:00 to 18:00 | |

Grassmann Matrices, Determinantal Assignment Problems and Approximate Decomposability N. Karcanias; J. Leventides Contact: Nicos Karcanias - United Kingdom | |

17:20-17:40 | |

Abstract: | |

The solution of exterior equations is an integral part of the study of the Determinantal Assignment Problem (DAP) of linear systems and its solvability (characterized by the property of multi-vector decomposability) is defined by the Quadratic Plücker Relations (QPR). An alternative new test for decomposability is given in this paper , in terms of the rank properties of a new form of matrices referred to as the Grassmann matrix of the corresponding multi-vector. It is shown that the exterior equation is solvable if and only if the right null space of the Grassmann matrix has given dimension. If this condition is satisfied, then the space associated with the exterior equation is defined as the null space of that matrix. The linear algebra formulation of the decomposability problem provides an alternative framework (to that defined by the QPRs) for the study of solvability and computation of solutions of DAP and enables the definition and study of “approximate solutions” of exterior equations as a distance problem. For the case of m=2, n=4 of exterior equations, the notion of approximate decomposability is introduced and a solution is given to the optimal approximation. The properties of the optimal approximate solution are linked to the singular values of the Grassmann matrix. Copyright © 2007 IFAC. |