The study of the invariants of homogeneous matrix polynomials using the extended hermite equivalence εrh

Grigoris I. Kalogeropoulos, Athanasios D. Karageorgos, Athanasios A. Pantelous

Research output: Chapter in Book/Report/Conference proceedingChapter (Book)Research


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), ℰp, defined on the setr of complex homogeneous binary polynomials of fixed homogeneous degree r. For a f (s,ŝ) ∈ r, the study of invariants of the PE class ℰp 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), ℰrh. 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 languageEnglish
Title of host publicationAdvances in Linear Algebra Research
PublisherNova Science Publishers
Number of pages22
ISBN (Electronic)9781634635806
ISBN (Print)9781634635653
Publication statusPublished - 1 Jan 2015
Externally publishedYes


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

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