The study of the invariants of homogeneous matrix polynomials using the extended hermite equivalence ε^rh

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_pis 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.