An Introduction To Semi-tensor Product Of Matrices And Its Applications

An Introduction To Semi-tensor Product Of Matrices And Its Applications

by Daizhan Cheng, Hongsheng Qi, Yin Zhao

Hardcover

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Overview

A generalization of Conventional Matrix Product (CMP), called the Semi-Tensor Product (STP), is proposed. It extends the CMP to two arbitrary matrices and maintains all fundamental properties of CMP. In addition, it has a pseudo-commutative property, which makes it more superior to CMP. The STP was proposed by the authors to deal with higher-dimensional data as well as multilinear mappings. After over a decade of development, STP has been proven to be a powerful tool in dealing with nonlinear and logical calculations.This book is a comprehensive introduction to the theory of STP and its various applications, including logical function, fuzzy control, Boolean networks, analysis and control of nonlinear systems, amongst others.

Product Details

ISBN-13: 9789814374682
Publisher: World Scientific Publishing Company, Incorporated
Publication date: 08/21/2012
Pages: 612
Product dimensions: 6.10(w) x 9.00(h) x 1.50(d)

Table of Contents

Preface v

Notations xiii

1 Multi-Dimensional Data 1

1.1 Multi-Dimensional Data 1

1.2 Arrangement of Data 4

1.3 Matrix Products 7

1.3.1 Kronecker Product of Matrices 8

1.3.2 Hadamard Product 9

1.3.3 Khatri-Rao Product 10

1.4 Tensor 11

1.5 Nash Equilibrium 14

1.6 Symmetric Group 16

1.7 Swap Matrix 18

Exercises 20

2 Semi-Tensor Product of Matrices 23

2.1 Multilinear Function 23

2.2 Left Semi-Tensor Product of Matrices 27

2.3 Fundamental Properties 32

2.4 Pseudo-Commutativity via Swap Matrix 38

2.5 Semi-Tensor Product as Bilinear Mapping 44

Exercises 47

3 Multilinear Mappings among Vector Spaces 51

3.1 Cross Product on R3 51

3.2 General Linear Algebra 55

3.3 Mappings over Matrices 58

3.4 Converting Matrix Expressions 69

3.5 Two Applications 75

3.5.1 General Linear Group and its Algebra 75

3.5.2 Hautus and Sylvester Equations 78

Exercises 80

4 Right and General Semi-Tensor Products 85

4.1 Right STP 85

4.2 Semi-Tensor Product of Arbitrary Matrices 92

Exercises 96

5 Rank, Pseudo-Inverse, and Positivity of STP 101

5.1 Rank of Products 101

5.2 Pseudo-Inverse of STP 102

5.2.1 Moore-Penrose Inverse 103

5.2.2 Drazin Inverse 107

5.3 Positivity of Products 108

Exercises 111

6 Matrix Expression of Logic 113

6.1 Logic and its Expression 113

6.2 General Structure of Logical Operators 121

6.3 Fundamental Properties of Logical Operators 124

6.4 Logical System and Logical Inference 130

6.5 Multi-Valued Logic 136

Exercises 141

7 Mix-Valued Logic 145

7.1 Normal Form of Logical Operators 145

7.2 Mix-Valued Logic 149

7.3 General Logical Mappings 152

7.4 Two Practical Examples 157

7.4.1 Mix-Valued Logical Form of Rules in Fuzzy Control 157

7.4.2 Expression of Strategies of Dynamic Games 160

Exercises 161

8 Logical Matrix, Fuzzy Set and Fuzzy Logic 163

8.1 Matrices of General Logical Variables 163

8.2 Logical Operators for k-Valued Matrices 166

8.3 Fuzzy Sets 169

8.4 Mappings over Fuzzy Sets 174

8.5 Fuzzy Logic and its Computation 178

Exercises 180

9 Fuzzy Relational Equation 185

9.1 k-Valued Matrix and Fuzzy Relational Equations 185

9.2 Structure of the Set of Solutions 188

9.3 Solving Fuzzy Relational Equation 191

9.4 Numerical Examples 193

9.5 Numerical Examples 193

Exercises 199

10 Fuzzy Control with Coupled Fuzzy Relations 201

10.1 Multiple Fuzzy Relations 201

10.1.1 Matrix Expression 201

10.1.2 Multiple Fuzzy Inference 203

10.1.3 Compounded Multiple Fuzzy Relations 204

10.2 Fuzzy Control of Coupled Multiple Fuzzy Relations 208

10.2.1 Fuzzification via Dual Fuzzy Structure 208

10.2.2 Design of Fuzzy Controller 210

10.2.3 Defuzzification 213

10.3 Numerical Solution for Fuzzy Control Design 216

Exercises 220

11 Representation of Boolean Functions 224

11.1 Boolean Functions in Galois Field Z2 225

11.2 Polynomial Form of Boolean Functions 228

11.3 Walsh Transformation 233

11.4 Linear Structure 240

11.5 Nonlinearity 244

11.6 Symmetry of Boolean Function 247

Exercises 251

12 Decomposition of Logical Functions 253

12.1 Disjoint Bi-Decomposition 253

12.2 Non-Disjoint Bi-Decomposition 260

12.3 Decomposition of Multi-Valued Logical Functions 265

12.4 Decomposition of Mix-Valued Logical Functions 269

Exercises 272

13 Boolean Calculus 275

13.1 Boolean Derivatives 275

13.2 Boolean Differential Equations 281

13.3 Boolean Integral 284

13.3.1 Primitive Function 285

13.3.2 Indefinite Integral 287

13.3.3 Definite Integral 292

Exercises 293

14 Lattice, Graph, and Universal Algebra 297

14.1 Lattice 297

14.2 Isomorphic Lattices and Sublattices 301

14.3 Matrix Expression of Finite Lattice 303

14.4 Distributive and Modular Lattices 309

14.5 Graph and its Adjacency Matrix 310

14.6 Vector Space Structure of Graph 314

14.7 Planar Graph and Coloring Problem 318

14.8 Universal Algebra 327

14.9 Lattice-Based Logics 332

Exercises 334

15 Boolean Network 337

15.1 An Introduction 337

15.2 Fixed Points and Cycles 340

15.3 Invariant Subspace and Input-State Description 343

15.3.1 State Space and Subspaces 343

15.3.2 Input-State Description 350

15.4 Higher-Order Boolean Networks 353

15.4.1 First Algebraic Form of Higher-Order Boolean Networks 354

15.4.2 Second Algebraic Form of Higher-Order Boolean Networks 356

15.5 Dynamic-Static Boolean Networks 357

Exercises 360

16 Boolean Control System 365

16.1 Dynamics of Boolean Control Networks 365

16.2 Controllability 370

16.3 Observability 373

16.4 Disturbance Decoupling 376

16.5 Some Other Control Problems 385

16.5.1 Stability and Stabilization 385

16.5.2 Optimal Control 391

16.5.3 Identification 393

Exercises 396

17 Game Theory 401

17.1 An Introduction to Game Theory 401

17.2 Infinitely Repeated Games 407

17.3 Local Optimization of Strategies and Local Nash/Sub-Nash Equilibrium 410

Exercises 415

18 Multi-Variable Polynomials 419

18.1 Matrix Expression of Multi-Variable Polynomials 419

18.2 Differential Form of Functional Matrices 430

18.3 Conversion of Generators 439

18.4 Taylor Expansion of Multi-Variable Functions 443

18.5 Fundamental Formula of Differential 447

18.6 Lie Derivative 449

Exercises 453

19 Some Applications to Differential Geometry and Algebra 457

19.1 Calculation of Connection 457

19.2 Contraction of Tensor Field 463

19.3 Structure Matrix of Finite-Dimensional Algebra 467

19.4 Two-Dimensional Algebras 473

19.5 Three-Dimensional Algebras 476

19.6 Lower-Dimensional Lie Algebra and Invertible Algebra 481

19.7 Tensor Product Algebra 492

Exercises 495

20 Morgan's Problem 499

20.1 Input-Output Decomposition 499

20.2 Problem Formulation 502

20.3 Numerical Expression of Solvability 505

Exercises 512

21 Linearization of Nonlinear Control Systems 515

21.1 Carleman Linearization 515

21.2 First Integral 519

21.3 Invariance of Polynomial System 525

21.4 Feedback Linearization of Nonlinear Control System 527

21.5 Single Input Feedback Linearization 529

21.6 Algorithm for Non-Regular Feedback Linearization 532

Exercises 537

22 Stability Region of Dynamic Systems 543

22.1 Stability Region 543

22.2 Stable Submanifold 545

22.3 Quadratic Approximation 548

22.4 Higher Order Approximation 552

22.5 Differential-Algebraic System 560

Exercises 563

Appendix A Numerical Algorithms 567

A.1 Basic Functions 567

A.2 Some Examples 570

Bibliography 573

Index 583

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