Positive Matrices

Abstract

The purpose of this introductory chapter is twofold. First, it supplies a unified discussion of the most important operator theoretic properties of positive square matrices; it can be read independently of the remainder of the book, presupposing only familiarity with elementary polynomial algebra (cf. § 1). Second, and more important in our context, this first chapter is intended to serve as a motivation for the study of positive operators on Banach lattices—the central theme of this book. As in the later chapters, the ideal structure of a vector lattice (cf. Chapter II, § 2) is used extensively and advantageously in Chapter I; even in the finite dimensional case discussed here, the judicious use of ideal theory turns out to be the key to many fruitful insights.