## Abstract

In systems and control theory, Linear Time Invariant (LTI) descriptor (Differential- Algebraic) systems are intimately related to the matrix pencil theory. Actually, a large number of systems are reduced to the study of differential (difference) systems S (F, G) of the form: S (FtG): Fx (t) = Gx(t) (or the dual Fx = Gx (t)0, and S (F, G): Fx_{k+1} = GX_{k} (or the dual Fx_{k} = Gx_{k+1}), F, G ε ℂ^{m×n} and their properties can be characterized by the homogeneous pencil ŝF - ŝG. An essential problem hi matrix pencil theory is the study of invariants of sF - ŝG under die bilinear strict equivalence. This problem is equivalent to the study of complete Projective Equivalence (PE), ε_{p}, defined on the set ℂ_{r} of complex homogeneous binaiy polynomials of fixed homogeneous degree r. For a / (s, s) ∈ ℂr, the study of invariants of the PE class ε_{p}is reduced to a study of invariants of matrices of the set ℂ^{k×2} (for k ≥ 3 with all 2 × 2-minors non-zero) under the Extended Hermite Equivalence (EHE), ε_{rh}. In this chapter, we present a review of the most interesting properties of the PE and the EHE classes. Moreover, the appropriate projective transformation d ∈ RGL(1,ℂ/ℝ) is provided analytically1.

Original language | English |
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Title of host publication | Mathematical Research Summaries |

Editors | Matthew A. Rowe |

Publisher | Nova Science Publishers |

Pages | 193-194 |

Number of pages | 2 |

Volume | 2 |

ISBN (Electronic) | 9781536122008 |

ISBN (Print) | 9781536120226 |

Publication status | Published - 1 Jan 2017 |

Externally published | Yes |