## 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: and their properties can be characterized by the homogeneous pencil sF - ŝG. An essential problem in matrix pencil theory is the study of invariants of sF - ŝG under the bilinear strict equivalence. This problem is equivalent to the study of complete Projective Equivalence (PE), ℰ_{p}, defined on the set_{r} of complex homogeneous binary polynomials of fixed homogeneous degree r. For a f (s,ŝ) ∈ _{r}, the study of invariants of the PE class ℰ_{p} is reduced to a study of invariants of matrices of the set ^{k×2 }(for k ≥ 3 with all 2 × 2-minors non-zero) under the Extended Hermite Equivalence (EHE), ℰ_{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, /ℝ) is provided analytically [1].

Original language | English |
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Title of host publication | Advances in Linear Algebra Research |

Publisher | Nova Science Publishers |

Pages | 57-78 |

Number of pages | 22 |

Edition | 1st |

ISBN (Electronic) | 9781634635806 |

ISBN (Print) | 9781634635653 |

Publication status | Published - 1 Jan 2015 |

Externally published | Yes |

## Keywords

- bilinear-strict equivalence
- elementary divisors
- extended Hermite equivalence
- linear systems
- matrix pencil theory