2. Analysis of Linear Systems

The goal of this chapter is to introduce two of the central topics in the book, i.e., symmetries and semi-invariants, for both continuous-time and discrete-time linear systems. The idea is that, being simpler to carry out computations in closed form for linear systems, this way the reader can become confident with such concepts before dealing with the unavoidable obstacles brought in by nonlinearities. The chapter is composed of two sections, the first one dealing with the concepts of centralizer and normalizer of a square matrix, and the second one dealing with first integrals and Darboux polynomials; Darboux polynomials are polynomial semi-invariants and, for linear systems, they constitute a significant subset of all the possible semi-invariants. A large portion of the first section is actually used to state and proof some important results of linear algebra that are central for the subsequent chapters.