Journal ArticleDOI
The Contraction Mapping Approach to the Perron-Frobenius Theory: Why Hilbert's Metric?
Elon Kohlberg,John W. Pratt +1 more
TLDR
The Perron-Frobenius Theorem says that if A is a nonnegative square matrix some power of which is positive, then there exists an x0 such that Anx/‖Anx‖ converges to xn for all x > 0.Abstract:
The Perron-Frobenius Theorem says that if A is a nonnegative square matrix some power of which is positive, then there exists an x0 such that Anx/‖Anx‖ converges to xn for all x > 0. There are many classical proofs of this theorem, all depending on a connection between positively of a matrix and properties of its eigenvalues. A more modern proof, due to Garrett Birkhoff, is based on the observation that every linear transformation with a positive matrix may be viewed as a contraction mapping on the nonnegative orthant. This observation turns the Perron-Frobenius theorem into a special ease of the Banach contraction mapping theorem. Furthermore, it applies equally to linear transformations which are positive in a much more general sense. The metric which Birkhoff used to show that positive linear transformations correspond to contraction mappings is known as Hilbert's projective metric. The definition of this metric is rather complicated. It is therefore natural to try to define another, less complicated m...read more
Citations
More filters
Journal ArticleDOI
Scalable Control of Positive Systems
TL;DR: In this paper, a method for synthesis of distributed controllers based on linear Lyapunov functions and storage functions instead of quadratic ones was developed for analysis and design of large scale control systems.
Journal ArticleDOI
Alternative modes of questioning in the analytic hierarchy process
TL;DR: In this paper, two extensions of the eigenvector approach of the Analytic Hierarchy Process (AHP) are presented, allowing the decision maker to say "I don't know" or "I'm not sure" to some of the questions being asked, and to approximate nonlinear functions of the ratios of the weights.
Book
Nonlinear Perron-Frobenius theory
Bas Lemmens,Roger D. Nussbaum +1 more
TL;DR: This is the first comprehensive and unified introduction to nonlinear Perron–Frobenius theory suitable for graduate students and researchers entering the field for the first time and acquaints the reader with recent developments and provides a guide to challenging open problems.
Journal ArticleDOI
On the scaling of multidimensional matrices
Joel Franklin,Jens Lorenz +1 more
TL;DR: For positive two-dimensional matrices, Hilbert's projective metric and a theorem of G. Birkhoff are used to prove that Sinkhorn's original iterative procedure converges geometrically; the ratio of convergence is estimated from the given data as discussed by the authors.
Journal ArticleDOI
The Perron-Frobenius theorem for homogeneous, monotone functions
TL;DR: In this article, the authors show that the Perron-Frobenius theorem is really about the boundedness of invariant subsets in the Hilbert projective metric, and show that if the graph is strongly connected, then f has a (nonlinear) eigenvector in (R + ) n.
References
More filters
Book
A first course in stochastic processes
Samuel Karlin,Howard M. Taylor +1 more
TL;DR: In this paper, the Basic Limit Theorem of Markov Chains and its applications are discussed and examples of continuous time Markov chains are presented. But they do not cover the application of continuous-time Markov chain in matrix analysis.
Book
Linear Topological Spaces
John L. Kelley,I. Namioka,William F. Donoghue,Kenneth R. Lucas,B. J. Pettis,Ebbe Thue Poulsen,G. Baley Price,Wendy Robertson,W. R. Scott,Kennan T. Smith +9 more
TL;DR: A brief review of some of the terminology and the elementary theorems of general topology, an examination of the new concept "linear topological space" in terms of more familiar notions, and a comparison of this new concept with the mathematical objects of which it is an abstraction can be found in this paper.
Journal ArticleDOI