Matrices and Matroids for Systems Analysis
A matroid is an abstract mathematical structure that captures combinatorial properties of matrices. This book offers a unique introduction to matroid theory, emphasizing motivations from matrix theory and applications to systems analysis. This book serves
- PDF / 4,836,582 Bytes
- 490 Pages / 439.37 x 666.142 pts Page_size
- 57 Downloads / 209 Views
Editorial Board R.L. Graham, La Jolla B. Korte, Bonn L. Lov´asz, Budapest A. Wigderson, Princeton G.M. Ziegler, Berlin
Kazuo Murota
Matrices and Matroids for Systems Analysis
123
Kazuo Murota Department of Mathematical Informatics Graduate School of Information Science and Technology University of Tokyo Tokyo, 113-8656 Japan [email protected]
ISBN 978-3-642-03993-5 e-ISBN 978-3-642-03994-2 DOI 10.1007/978-3-642-03994-2 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2009937412 c Springer-Verlag Berlin Heidelberg 2000, first corrected softcover printing 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: deblik, Berlin Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface
Interplay between matrix theory and matroid theory is the main theme of this book, which offers a matroid-theoretic approach to linear algebra and, reciprocally, a linear-algebraic approach to matroid theory. The book serves also as the first comprehensive presentation of the theory and application of mixed matrices and mixed polynomial matrices. A matroid is an abstract mathematical structure that captures combinatorial properties of matrices, and combinatorial properties of matrices, in turn, can be stated and analyzed successfully with the aid of matroid theory. The most important result in matroid theory, deepest in mathematical content and most useful in application, is the intersection theorem, a duality theorem for a pair of matroids. Similarly, combinatorial properties of polynomial matrices can be formulated in the language of valuated matroids, and moreover, the intersection theorem can be generalized for a pair of valuated matroids. The concept of a mixed matrix was formulated in the early eighties as a mathematical tool for systems analysis by means of matroid-theoretic combinatorial methods. A matrix is called a mixed matrix if it is expressed as the sum of a “constant” matrix and a “generic” matrix having algebraically independent nonzero entries. This concept is motivated by the physical observation that two different kinds of numbers, fixed constants and system parameters, are to be distinguished in the description of engineeri
Data Loading...
