Rings and Modules of Polynomials

Abstract

Since concepts from algebra such as rings and modules play a dominant role in our polynomial approach to linear systems, it seems useful to collect in this chapter some of the most relevant concepts and results from linear algebra. We assume that the reader is already familiar with the basic definitions and facts of linear algebra over a field. Starting from such a background, our aim is to generalize the well-known theory of vector spaces and linear operators over a field to an analogous theory of modules, defined over the ring of polynomials in one variable. From a system-theoretic point of view this corresponds to passing from a state-space point of view to a frequency-domain description of linear systems, and even further on to a module-theoretic approach. This algebraic approach to linear systems theory was first expounded in the very influential book by Kalman, Falb and Arbib (1969) and has led to a new area in applied mathematics called algebraic systems theory. It underpins our work on the functional model approach to linear systems, a theory that forms the mathematical basis of this book. In doing so it proves important to be able to extend the classical arithmetic properties of scalar polynomials, such as divisibility, coprimeness, and the Chinese remainder theorem, to rectangular polynomial matrices. This is done in the subsequent sections of this chapter. Key facts are the Hermite and Smith canonical forms for polynomial matrices and rational matrices, respectively, Wiener–Hopf factorizations, and the basic existence and uniqueness properties for left and right coprime factorizations of rational matrix-valued functions. We study doubly coprime factorizations that are associated with intertwining relations between pairs of left and right coprime polynomial matrices. The existence of such doubly coprime factorizations will play a crucial role later on in our approach to open-loop control.