What is nonlinear Perron-Frobenius theory? To get an impression of the contents of nonlinear Perron-Frobenius theory, it is useful to first recall the basics of classical Perron-Frobenius theory. Classical Perron-Frobenius theory concerns nonnegative matrices, their eigenvalues and corresponding eigenvectors. The fundamental theorems of this classical theory were discovered at the beginning of the twentieth century by Perron [179, 180], who investigated eigenvalues and eigenvectors of matrices with strictly positive entries, and by Frobenius [70-72], who extended Perron's results to irreducible nonnegative matrices. In the first section we discuss the theorems of Perron and Frobenius and some of their generalizations to linear maps that leave a cone in a finite-dimensional vector space invariant. The proofs of these classical results can be found in many books on matrix analysis, e.g., [15, 22, 73, 148, 202]. Nevertheless, in Appendix B we prove some of them once more using a combination of analytic, geometric, and algebraic methods. The geometric methods originate from work of Alexandroff and Hopf [8], Birkhoff [25], Kreȋn and Rutman [117], and Samelson [192] and underpin much of nonlinear Perron-Frobenius theory. Readers who are not familiar with these methods might prefer to first read Chapters 1 and 2 and Appendix B. Besides recalling the classical Perron-Frobenius theorems, we use this chapter to introduce some basic concepts and terminology that will be used throughout the exposition, and provide some motivating examples of classes of nonlinear maps to which the theory applies. We emphasize that throughout the book we will always be working in a finite-dimensional real vector space V , unless we explicitly say otherwise. 1.1 Classical Perron-Frobenius theory An n × n matrix A = (a i j) is said to be nonnegative if a i j ≥ 0 for all i and j. It is called positive if a i j > 0 for all i and j. Similarly, we call a vector x ∈ R n www.cambridge.org © in this web service Cambridge University Press Cambridge University Press 978-0-521-89881-2-Nonlinear Perron-Frobenius Theory Bas Lemmens and Roger Nussbaum Excerpt More information 2 What is nonlinear Perron-Frobenius theory? nonnegative (or positive) if all its coordinates are nonnegative (or positive). The spectrum of A is given by σ (A) = {λ ∈ C : Ax = λx for some x ∈ C n \ {0}}. Recall also that the spectral radius of A is given by r (A) = max{|λ| : λ ∈ σ (A)}, and satisfies the equality r (A) = lim k→∞ A k 1/k. Notice that the limit is independent of the choice of the matrix norm, or norm on R n 2 , as they are all equivalent; see Rudin [190]. The following result is due to Perron [180]. Theorem 1.1.2 (Perron-Frobenius) If A is a nonnegative irreducible n × n matrix, then the following assertions hold:
