Infinite elementary divisors of a matrix polynomial and implications

Equivalence of AR‐representations in the light of the impulsive‐smooth behaviour

“…We should note that the notion of fundamental equivalence proposed here ensures the smooth-impulsive behaviours of the two systems are identical. We establish that the matrix conditions guaranteeing fundamental equivalence coincide with those of full unimodular equivalence (presented in Section 3), which is a special case of full matrix equivalence presented in [19]. This provides a natural connection between the behavioural setting and the theory of polynomial matrix transformations.…”

Construction of Algebraic and Difference Equations with a Prescribed Solution Space

“…The connection between the Smith form at the infinity of A(σ) and the Smith form at the zero of the dual matrix is given by Hayton et al (1988) and Vardulakis (1991): …”

“…Antoniou et al, 1998), whereas the results were extended for non-regular systems by Karampetakis (2004). The algebraic structure of polynomial matrices was studied by Bernstein (2009), Gantmacher (1959), Gohberg et al (2009), Hayton et al (1988), Kaczorek (2007), Karampetakis and Vologiannidis (2003), Karampetakis et al (2004), Praagman (1991), Zaballa and Tisseur (2012), and in the references cited therein. An interesting problem that we face in this paper is the following inverse problem: Given a certain forward/backward solution space, find a system of algebraic and difference equations with the prescribed solution space.…”

Spectral structures of the generalized companion form and applications

“…2 It is noted that the matrix sI n − J contains the finite elementary divisors of C A (s), and the matrix sJ ∞ − I ν contains the infinite elementary divisors [10] of C A (s). It is also noted in [6] that the matrix pencil C A (s) has the same finite elementary divisors as A(s).…”

“…The infinite frequency structure of A(s) is related to the finite frequency structure of its dual D A (w) at w = 0 in the manner described in [10]…”