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: 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].