Matrices and Matroids for System Analysis. Kazuo Murota.pdf...

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Unformatted text preview: Algorithms and Combinatorics Volume 20 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 ( ) 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 engineering systems. Mathematical analysis of a mixed matrix can be streamlined by the intersection theorem applied to the pair of matroids associated with the “constant” and “generic” matrices. This approach can be extended further to a mixed polynomial matrix on the basis of the intersection theorem for valuated matroids. The present volume grew out of an attempted revision of my previous monograph, “Systems Analysis by Graphs and Matroids — Structural Solvability and Controllability” (Algorithms and Combinatorics, Vol. 3, SpringerVerlag, Berlin, 1987), which was an improved presentation of my doctoral thesis written in 1983. It was realized, however, that the progress made in the last decade was so remarkable that even a major revision was inadequate. The present volume, sharing the same approach initiated in the above mono- VI Preface graph, offers more advanced results obtained since then. For developments in the neighboring areas the reader is encouraged to consult: • A. Recski: “Matroid Theory and Its Applications in Electric Network Theory and in Statics” (Algorithms and Combinatorics, Vol. 6, SpringerVerlag, Berlin, 1989), • R. A. Brualdi and H. J. Ryser: “Combinatorial Matrix Theory” (Encyclopedia of Mathematics and Its Applications, Vol. 39, Cambridge University Press, London, 1991), • H. Narayanan: “Submodular Functions and Electrical Networks” (Annals of Discrete Mathematics, Vol. 54, Elsevier, Amsterdam, 1997). The present book is intended to be read profitably by graduate students in engineering, mathematics, and computer science, and also by mathematicsoriented engineers and application-oriented mathematicians. Self-contained presentation is envisaged. In particular, no familiarity with matroid theory is assumed. Instead, the book is written in the hope that the reader will acquire familiarity with matroids through matrices, which should certainly be more familiar to the majority of the readers. Abstract theory is always accompanied by small examples of concrete matrices. Chapter 1 is a brief introduction to the central ideas of our combinatorial method for the structural analysis of engineering systems. Emphasis is laid on relevant physical observations that are crucial to successful mathematical modeling for structural analysis. Chapter 2 explains fundamental facts about matrices, graphs, and matroids. A decomposition principle based on submodularity is described and the Dulmage–Mendelsohn decomposition is derived as its application. Chapter 3 discusses the physical motivation of the concepts of mixed matrix and mixed polynomial matrix. The dual viewpoint from structural analysis and dimensional analysis is explained by way of examples. Chapter 4 develops the theory of mixed matrices. Particular emphasis is put on the combinatorial canonical form (CCF) of layered mixed matrices and related decompositions, which generalize the Dulmage–Mendelsohn decomposition. Applications to the structural solvability of systems of equations are also discussed. Chapter 5 is mostly devoted to an exposition of the theory of valuated matroids, preceded by a concise account of canonical forms of polynomial/rational matrices. Chapter 6 investigates mathematical properties of mixed polynomial matrices using the CCF and valuated matroids as main tools of analysis. Control theoretic problems are treated by means of mixed polynomial matrices. Chapter 7 presents three supplementary topics: the combinatorial relaxation algorithm, combinatorial system theory, and mixed skew-symmetric matrices. Expressions are referred to by their numbers; for example, (2.1) designates the expression (2.1), which is the first numbered expression in Chap. 2. Preface VII Similarly for figures and tables. Major symbols used in this book are listed in Notation Table. The ideas and results presented in this book have been developed with the help, guidance, encouragement, support, and criticisms offered by many people. My deepest gratitude is expressed to Professor Masao Iri, who introduced me to the field of mathematical engineering and guided me as the thesis supervisor. I appreciate the generous hospitality of Professor Bernhard Korte during my repeated stays at the University of Bonn, where a considerable part of the theoretical development was done. I benefited substantially from discussions and collaborations with Pawel Bujakiewicz, Fran¸cois Cellier, Andreas Dress, Jim Geelen, Andr´as Frank, Hisashi Ito, Satoru Iwata, Andr´ as Recski, Mark Scharbrodt, Andr´ as Seb˝ o, Masaaki Sugihara, and Jacob van der Woude. Several friends helped me in writing this book. Most notable among these were Akiyoshi Shioura and Akihisa Tamura who went through all the text and provided comments. I am also indebted to Daisuke Furihata, Koichi Kubota, Tomomi Matsui, and Reiko Tanaka. Finally, I thank the editors of Springer-Verlag, Joachim Heinze and Martin Peters, for their support in the production of this book, and Erich Goldstein for English editing. Kyoto, June 1999 Kazuo Murota VIII Preface Preface to the Softcover Edition Since the appearance of the original edition in 2000 steady progress has been made in the theory and application of mixed matrices. Geelen–Iwata [354] gives a novel rank formula for mixed skew-symmetric matrices and derives therefrom the Lov´ asz min-max formula in Remark 7.3.2 for the linear matroid parity problem. Harvey–Karger–Murota [355] and Harvey–Karger–Yekhanin [356] exploit mixed matrices in the context of matrix completion; the former discussing its application to network coding. Iwata [357] proposes a matroidal abstraction of matrix pencils and gives an alternative proof for Theorem 7.2.11. Iwata–Shimizu [358] discusses a combinatorial characterization for the singular part of the Kronecker form of generic matrix pencils, extending the graph-theoretic characterization for regular pencils by Theorem 5.1.8. Iwata– Takamatsu [359] gives an efficient algorithm for computing the degrees of all cofactors of a mixed polynomial matrix, a nice combination of the algorithm of Section 6.2 with the all-pair shortest path algorithm. Iwata–Takamatsu [360] considers minimizing the DAE index, in the sense of Section 1.1.1, in hybrid analysis for circuit simulation, giving an efficient solution algorithm by making use of the algorithm [359] above. In the softcover edition, updates and corrections are made in the reference list: [59], [62], [82], [91], [93], [139], [141], [142], [146], [189], [236], [299], [327]. References [354] to [360] mentioned above are added. Typographical errors in the original edition have been corrected: MQ is changed to M(Q) in lines 26 and 34 of page 142, and ∂(M ∩ CQ ) is changed to ∂M ∩ CQ in line 12 of page 143 and line 5 of page 144. Tokyo, July 2009 Kazuo Murota Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. 2. V Introduction to Structural Approach — Overview of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Structural Approach to Index of DAE . . . . . . . . . . . . . . . . . . . . . 1.1.1 Index of Differential-algebraic Equations . . . . . . . . . . . . 1.1.2 Graph-theoretic Structural Approach . . . . . . . . . . . . . . . 1.1.3 An Embarrassing Phenomenon . . . . . . . . . . . . . . . . . . . . . 1.2 What Is Combinatorial Structure? . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Two Kinds of Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Descriptor Form Rather than Standard Form . . . . . . . . 1.2.3 Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Mathematics on Mixed Polynomial Matrices . . . . . . . . . . . . . . . 1.3.1 Formal Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Resolution of the Index Problem . . . . . . . . . . . . . . . . . . . 1.3.3 Block-triangular Decomposition . . . . . . . . . . . . . . . . . . . . 1 1 1 3 7 10 11 15 17 20 20 21 26 Matrix, Graph, and Matroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Polynomial and Algebraic Independence . . . . . . . . . . . . . 2.1.2 Determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Rank, Term-rank and Generic-rank . . . . . . . . . . . . . . . . . 2.1.4 Block-triangular Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Directed Graph and Bipartite Graph . . . . . . . . . . . . . . . . 2.2.2 Jordan–H¨ older-type Theorem for Submodular Functions 2.2.3 Dulmage–Mendelsohn Decomposition . . . . . . . . . . . . . . . 2.2.4 Maximum Flow and Menger-type Linking . . . . . . . . . . . 2.2.5 Minimum Cost Flow and Weighted Matching . . . . . . . . 2.3 Matroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 From Matrix to Matroid . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Basis Exchange Properties . . . . . . . . . . . . . . . . . . . . . . . . . 31 31 31 33 36 40 43 43 48 55 65 67 71 71 73 77 78 X Contents 2.3.5 Independent Matching Problem . . . . . . . . . . . . . . . . . . . . 84 2.3.6 Union . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 2.3.7 Bimatroid (Linking System) . . . . . . . . . . . . . . . . . . . . . . . 97 3. Physical Observations for Mixed Matrix Formulation . . . . . 3.1 Mixed Matrix for Modeling Two Kinds of Numbers . . . . . . . . . 3.1.1 Two Kinds of Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Mixed Matrix and Mixed Polynomial Matrix . . . . . . . . . 3.2 Algebraic Implication of Dimensional Consistency . . . . . . . . . . 3.2.1 Introductory Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Dimensioned Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Total Unimodularity of a Dimensioned Matrix . . . . . . . 3.3 Physical Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Physical Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Physical Matrices in a Dynamical System . . . . . . . . . . . 107 107 107 116 120 120 121 123 126 126 128 4. Theory and Application of Mixed Matrices . . . . . . . . . . . . . . . 4.1 Mixed Matrix and Layered Mixed Matrix . . . . . . . . . . . . . . . . . . 4.2 Rank of Mixed Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Rank Identities for LM-matrices . . . . . . . . . . . . . . . . . . . . 4.2.2 Rank Identities for Mixed Matrices . . . . . . . . . . . . . . . . . 4.2.3 Reduction to Independent Matching Problems . . . . . . . 4.2.4 Algorithms for the Rank . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Structural Solvability of Systems of Equations . . . . . . . . . . . . . . 4.3.1 Formulation of Structural Solvability . . . . . . . . . . . . . . . . 4.3.2 Graphical Conditions for Structural Solvability . . . . . . . 4.3.3 Matroidal Conditions for Structural Solvability . . . . . . . 4.4 Combinatorial Canonical Form of LM-matrices . . . . . . . . . . . . . 4.4.1 LM-equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Theorem of CCF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Construction of CCF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.4 Algorithm for CCF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.5 Decomposition of Systems of Equations by CCF . . . . . . 4.4.6 Application of CCF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.7 CCF over Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Irreducibility of LM-matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Theorems on LM-irreducibility . . . . . . . . . . . . . . . . . . . . . 4.5.2 Proof of the Irreducibility of Determinant . . . . . . . . . . . 4.6 Decomposition of Mixed Matrices . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 LU-decomposition of Invertible Mixed Matrices . . . . . . 4.6.2 Block-triangularization of General Mixed Matrices . . . . 4.7 Related Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.1 Decomposition as Matroid Union . . . . . . . . . . . . . . . . . . . 4.7.2 Multilayered Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.3 Electrical Network with Admittance Expression . . . . . . 131 131 134 135 139 142 145 153 153 156 160 167 167 172 175 181 187 191 199 202 202 205 211 212 215 221 221 225 228 Contents XI 4.8 Partitioned Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.2 Existence of Proper Block-triangularization . . . . . . . . . . 4.8.3 Partial Order Among Blocks . . . . . . . . . . . . . . . . . . . . . . . 4.8.4 Generic Partitioned Matrix . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Principal Structures of LM-matrices . . . . . . . . . . . . . . . . . . . . . . 4.9.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9.2 Principal Structure of Submodular Systems . . . . . . . . . . 4.9.3 Principal Structure of Generic Matrices . . . . . . . . . . . . . 4.9.4 Vertical Principal Structure of LM-matrices . . . . . . . . . . 4.9.5 Horizontal Principal Structure of LM-matrices . . . . . . . 230 231 235 238 240 250 250 252 254 257 261 5. Polynomial Matrix and Valuated Matroid . . . . . . . . . . . . . . . . 5.1 Polynomial/Rational Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Polynomial Matrix and Smith Form . . . . . . . . . . . . . . . . 5.1.2 Rational Matrix and Smith–McMillan Form at Infinity 5.1.3 Matrix Pencil and Kronecker Form . . . . . . . . . . . . . . . . . 5.2 Valuated Matroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Basic Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Greedy Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.5 Valuated Bimatroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.6 Induction Through Bipartite Graphs . . . . . . . . . . . . . . . . 5.2.7 Characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.8 Further Exchange Properties . . . . . . . . . . . . . . . . . . . . . . . 5.2.9 Valuated Independent Assignment Problem . . . . . . . . . . 5.2.10 Optimality Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.11 Application to Triple Matrix Product . . . . . . . . . . . . . . . 5.2.12 Cycle-canceling Algorithms . . . . . . . . . . . . . . . . . . . . . . . . 5.2.13 Augmenting Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 271 271 272 275 280 280 281 282 285 287 290 295 300 306 308 316 317 325 6. Theory and Application of Mixed Polynomial Matrices . . . 6.1 Descriptions of Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Mixed Polynomial Matrix Descriptions . . . . . . . . . . . . . . 6.1.2 Relationship to Other Descriptions . . . . . . . . . . . . . . . . . 6.2 Degree of Determinant of Mixed Polynomial Matrices . . . . . . . 6.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Graph-theoretic Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Basic Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 Reduction to Valuated Independent Assignment . . . . . . 6.2.5 Duality Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.6 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Smith Form of Mixed Polynomial Matrices . . . . . . . . . . . . . . . . 6.3.1 Expression of Invariant Factors . . . . . . . . . . . . . . . . . . . . . 331 331 331 332 335 335 336 337 340 343 348 355 355 XII 7. Contents 6.3.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Controllability of Dynamical Systems . . . . . . . . . . . . . . . . . . . . . 6.4.1 Controllability . . . . . . . . . . . . . . . . . . . . ....
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