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): Fxk+1 = GXk (or the dual Fxk = Gxk+1), F, G ε ℂm×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), εp, defined on the set ℂr of complex homogeneous binaiy polynomials of fixed homogeneous degree r. For a / (s, s) ∈ ℂr, the study of invariants of the PE class εpis reduced to a study of invariants of matrices of the set ℂk×2 (for k ≥ 3 with all 2 × 2-minors non-zero) under the Extended Hermite Equivalence (EHE), εrh. 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.
|Title of host publication||Mathematical Research Summaries|
|Editors||Matthew A. Rowe|
|Publisher||Nova Science Publishers|
|Number of pages||2|
|Publication status||Published - 1 Jan 2017|