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 also as a comprehensive presentation of the theory and application of mixed matrices, developed primarily by the present author in the last decade. A mixed matrix is a convenient mathematical tool for systems analysis, compatible with the physical observation that "fixed constants" and "system parameters" are to be distinguished in the description of engineering systems. This book will be extremely useful to graduate students and researchers in engineering, mathematics and computer science.
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#4408
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#4408
Read It:
Yes
Matrices, Matroids, System analysis
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Notes
Preface I. Introduction to Structural Approach
Overview of the Book
1 Structural Approach to Index of DAE
1.1 Index of differential-algebraic equations
1.2 Graph-theoretic structural approach
1.3 An embarrassing phenomenon
2 What Is Combinatorial Structure?
2.1 Two kinds of numbers
2.2 Descriptor form rather than standard form
2.3 Dimensional analysis
3 Mathematics on Mixed Polynomial Matrices
3.1 Formal definitions
3.2 Resolution of the index problem
3.3 Block-triangular decomposition
II. Matrix, Graph and Matroid
4 Matrix
4.1 Polynomial and algebraic independence
4.2 Determinant
4.3 Rank, term-rank and generic-rank
4.4 Block-triangular forms
5 Graph
5.1 Directed graph and bipartite graph
5.2 Jordan-Holder-type theorem for submodular functions
5.3 Dulmage-Mendelsohn decomposition
5.4 Maximum flow and Menger-type linking
5.5 Minimum cost flow and weighted matching
6 Matroid
6.1 From matrix to matroid
6.2 Basic concepts
6.3 Examples
6.4 Basis exchange properties
6.5 Independent matching problem
6.6 Union
6.7 Bimatroid (linking system)
III. Physical Observations for Mixed Matrix Formulation
7 Mixed Matrix for Modeling Two Kinds of Numbers
7.1 Two kinds of numbers
7.2 Mixed matrix and mixed polynomial matrix
8 Algebraic Implications of Dimensional Consistency
8.1 Introductory comments
8.2 Dimensioned matrix
8.3 Total unimodularity of dimensioned matrices
9 Physical Matrix
9.1 Physical matrix
9.2 Physical matrices in a dynamical system
IV. Theory and Application of Mixed Matrices
10 Mixed Matrix and Layered Mixed Matrix
11 Rank of Mixed Matrices
11.1 Rank identities for LM-matrices
11.2 Rank identities for mixed matrices
11.3 Reduction to independent matching problems
11.4 Algorithms for the rank
11.4.1 Algorithm for LM-matrices
11.4.2 Algorithm for mixed matrices
12 Structural Solvability of Systems of Equations
12.1 Formulation of structural solvability
12.2 Graphical conditions for structural solvability
12.3 Matroidal conditions for structural solvability
13. Co
Overview of the Book
1 Structural Approach to Index of DAE
1.1 Index of differential-algebraic equations
1.2 Graph-theoretic structural approach
1.3 An embarrassing phenomenon
2 What Is Combinatorial Structure?
2.1 Two kinds of numbers
2.2 Descriptor form rather than standard form
2.3 Dimensional analysis
3 Mathematics on Mixed Polynomial Matrices
3.1 Formal definitions
3.2 Resolution of the index problem
3.3 Block-triangular decomposition
II. Matrix, Graph and Matroid
4 Matrix
4.1 Polynomial and algebraic independence
4.2 Determinant
4.3 Rank, term-rank and generic-rank
4.4 Block-triangular forms
5 Graph
5.1 Directed graph and bipartite graph
5.2 Jordan-Holder-type theorem for submodular functions
5.3 Dulmage-Mendelsohn decomposition
5.4 Maximum flow and Menger-type linking
5.5 Minimum cost flow and weighted matching
6 Matroid
6.1 From matrix to matroid
6.2 Basic concepts
6.3 Examples
6.4 Basis exchange properties
6.5 Independent matching problem
6.6 Union
6.7 Bimatroid (linking system)
III. Physical Observations for Mixed Matrix Formulation
7 Mixed Matrix for Modeling Two Kinds of Numbers
7.1 Two kinds of numbers
7.2 Mixed matrix and mixed polynomial matrix
8 Algebraic Implications of Dimensional Consistency
8.1 Introductory comments
8.2 Dimensioned matrix
8.3 Total unimodularity of dimensioned matrices
9 Physical Matrix
9.1 Physical matrix
9.2 Physical matrices in a dynamical system
IV. Theory and Application of Mixed Matrices
10 Mixed Matrix and Layered Mixed Matrix
11 Rank of Mixed Matrices
11.1 Rank identities for LM-matrices
11.2 Rank identities for mixed matrices
11.3 Reduction to independent matching problems
11.4 Algorithms for the rank
11.4.1 Algorithm for LM-matrices
11.4.2 Algorithm for mixed matrices
12 Structural Solvability of Systems of Equations
12.1 Formulation of structural solvability
12.2 Graphical conditions for structural solvability
12.3 Matroidal conditions for structural solvability
13. Co