Handbook of Linear Algebra, 2nd Edition

Chapter 36

Matrices Leaving a Cone Invariant

Bit-Shun Tam

Tamkang University

Hans Schneider

University of Wisconsin-Madison

Generalizations of the Perron–Frobenius theory of nonnegative matrices to linear operators leaving a cone invariant were first developed for operators on a Banach space by Krein and Rutman [KR48], Karlin [Kar59], and Schaefer [Sfr66], although there are early examples in finite dimensions, e.g., [Sch65] and [Bir67]. In this chapter, we describe a generalization that is sometimes called the geometric spectral theory of nonnegative linear operators in finite dimensions, which emerged in the late 1980s. Motivated by a search for geometric analogs of results in the previously developed combinatorial spectral theory of (reducible) ...