Nonnegative matrices and applications / R.B. Bapat, T.E.S. Raghavan.

Format:

Book

Author/Creator:

Bapat, R. B., author.

Raghavan, T. E. S., author.

Series:

Encyclopedia of mathematics and its applications ; v. 64.

Encyclopedia of mathematics and its applications ; volume 64

Language:

English

Subjects (All):

Non-negative matrices.

Physical Description:

1 online resource (xiii, 336 pages) : digital, PDF file(s).

Other Title:

Nonnegative Matrices & Applications

Place of Publication:

Cambridge : Cambridge University Press, 1997.

Language Note:

English

Summary:

This book provides an integrated treatment of the theory of nonnegative matrices (matrices with only positive numbers or zero as entries) and some related classes of positive matrices, concentrating on connections with game theory, combinatorics, inequalities, optimisation and mathematical economics. The wide variety of applications, which include price fixing, scheduling and the fair division problem, have been carefully chosen both for their elegant mathematical content and for their accessibility to students with minimal preparation. Many results in matrix theory are also presented. The treatment is rigorous and almost all results are proved completely. These results and applications will be of great interest to researchers in linear programming, statistics and operations research. The minimal prerequisites also make the book accessible to first-year graduate students.

Contents:

Cover; Title; Copyright; Dedication; Contents; Preface; 1 Perron-Frobenius theory and matrix games; 1.1 Irreducible nonnegative matrices; 1.2 Perron's Theorem on positive matrices; 1.3 Completely mixed games; 1.4 The Perron-Frobenius theorem; 1.5 Nonsingular M-matrices; 1.6 Polyhedral sets with least elements; 1.7 Reducible nonnegative matrices; 1.8 Primitive matrices; 1.9 Finite Markov chains; 1.10 Self maps of the Lorentz cone; Exercises; 2 Doubly stochastic matrices; 2.1 The Birkhoff-von Neumann Theorem; 2.2 Fully indecomposable matrices; 2.3. König's Theorem and rank

2.4 The optimal assignment problem2.5 A probabilistic algorithm; 2.6 Diagonal products; 2.7 A self map of doubly stochastic matrices; 2.8 van der Waerden conjecture and its solution; 2.9 Cooperative games with side payments; 2.10. Lexicographic center; 2.11 Open shop scheduling; 2.12 A fair division problem; Exercises; 3 Inequalities; 3.1 Perron root and row sums; 3.2 Applications of the Information Inequality; 3.3 Inequalities of Levinger and Kingman; 3.4 Sum-symmetric matrices; 3.5 Circuit geometric means; 3.6 The Hadamard Inequality; 3.7 Inequalities of Fiedler and Oppenheim

3.8 Schur power matrix3.9 Majorization inequalities for eigenvalues; 3.10 The parallel sum; 3.11 Symmetric function means; Exercises; 4 Conditionally positive definite matrices; 4.1 Distance matrices; 4.2 Quasi-convex quadratic forms; 4.3 An interpolation problem; 4.4 A characterization theorem; 4.5 Log-concavity and discrete distributions; 4.6. The q-permanent; Exercises; 5 Topics in combinatorial theory; 5.1 Matroids; 5.2 Mixed discriminants; 5.3 The Alexandroff Inequality; 5.4 Coxeter graphs; 5.5 Matrices over the max algebra; 5.6 Boolean matrices; Exercises

6 Scaling problems and their applications6.1 Practical examples of scaling problems; 6.2 Kronecker Index Theorem and scaling; 6.3 Hilbert's projective metric; 6.4 Algorithms for scaling; 6.5 Maximum likelihood estimation; Exercises; 7 Special matrices in economic models; 7.1 Pure exchange economy; 7.2 Linear slave economies; 7.3 Substitution Theorem; 7.4 Sraffa system; 7.5 Dual Sraffa system on quantities; 7.6 A linear model of an expanding economy; 7.7 Factor price equalization; 7.8 P-matrices; 7.9 N-matrices; 7.10 Global univalence; 7.11 Stability and market prices; 7.12 Historical notes

ExercisesReferences; Index; Author Index

Notes:

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

Includes bibliographical references (p. 315-328) and index.

ISBN:

1-139-88668-1

0-511-96786-1

1-107-10328-2

0-521-11866-2

0-511-83574-4

1-107-09513-1

1-107-08894-1

0-511-52997-X

1-107-09187-X

OCLC:

862939767