Rational matrices and vector spaces

Abstract

In this section we summarize some basic algebraic definitions and results that will be needed throughout this book. For more details the reader is referred to textbooks on matrices over rings such as [47]. We assume that the reader is familiar with the notion of a field as a collection of objects that can be added, subtracted, multiplied and divided with the usual associative, distributive and commutative rules. The fields of importance in this book are the field of real numbers ℝ, the field of complex numbers ℂ and the field of rational functions in s with coefficients in ℂ (ℝ), written conventionally as ℂ(s) (ℝ(s)). We also assume that the reader is familiar with the notion of a commutative ring as a collection of objects with all the properties of a field except division. The set of polynomials in s with coefficients in ℂ (or ℝ) is a commutative ring, written conventionally as ℂ[s] (ℝ[s]).