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.

Description-Table Of Contents

1. Multi-dimensional data. 1. Multi-dimensional data. 1.2. Arrangement of data. 1.3. Matrix products. 1.4. Tensor. 1.5. Nash equilibrium. 1.6. Symmetric group. 1.7. Swap matrix -- 2. Semi-tensor product of matrices. 2.1. Multilinear function. 2.2. Left semi-tensor product of matrices. 2.3. Fundamental properties. 2.4. Pseudo-commutativity via swap matrix. 2.5. Semi-tensor product as bilinear mapping -- 3. Multilinear Mappings among vector spaces. 3.1. Cross product on [symbol]. 3.2. General linear algebra. 3.3. Mappings over matrices. 3.4. Converting matrix expressions. 3.5. Two applications -- 4. Right and general semi-tensor products. 4.1. Right STP. 4.2. Semi-tensor product of arbitrary matrices -- 5. Rank, pseudo-inverse, and positivity of STP. 5.1. Rank of products. 5.2. Pseudo-inverse of STP. 5.3. Positivity of products -- 6. Matrix expression of logic. 6.1. Logic and its expression. 6.2. General structure of logical operators. 6.3. Fundamental properties of logical operators. 6.4. Logical system and logical inference. 6.5. Multi-valued logic -- 7. Mix-valued logic. 7.1. Normal form of logical operators. 7.2. Mix-valued logic. 7.3. General logical mappings. 7.4. Two practical examples -- 8. Logical matrix, fuzzy set and fuzzy logic. 8.1. Matrices of general logical variables. 8.2. Logical operators for k-valued matrices. 8.3. Fuzzy sets. 8.4. Mappings over fuzzy sets. 8.5. Fuzzy logic and its computation -- 9. Fuzzy relational equation. 9.1. k-valued matrix and fuzzy relational equations. 9.2. Structure of the set of solutions. 9.3. Solving fuzzy relational equation. 9.4. Numerical examples -- 10. Fuzzy control with coupled fuzzy relations. 10.1. Multiple fuzzy relations. 10.2. Fuzzy control of coupled multiple fuzzy relations. 10.3. Numerical solution for fuzzy control design -- 11. Representation of boolean functions. 11.1. Boolean functions in Galois field [symbol]. 11.2. Polynomial form of boolean functions. 11.3. Walsh transformation. 11.4. Linear structure. 11.5. Nonlinearity. 11.6. Symmetry of boolean function. ; 8 12. Decomposition of logical functions. 12.1. Disjoint bi-decomposition. 12.2. Non-disjoint bi-decomposition. 12.3. Decomposition of multi-valued logical functions. 12.4. Decomposition of mix-valued logical functions -- 13. Boolean calculus. 13.1. Boolean derivatives. 13.2. Boolean differential equations. 13.3. Boolean integral -- 14. Lattice, graph, and universal algebra. 14.1. Lattice. 14.2. Isomorphic lattices and sublattices. 14.3. Matrix expression of finite lattice. 14.4. Distributive and modular lattices. 14.5. Graph and its adjacency matrix. 14.6. Vector space structure of graph. 14.7. Planar graph and coloring problem. 14.8. Universal algebra. 14.9. Lattice-based logics -- 15. Boolean network. 15.1. An introduction. 15.2. Fixed points and cycles. 15.3. Invariant subspace and input-state description. 15.4. Higher-order boolean networks. 15.5. Dynamic-static boolean networks -- 16. Boolean control system. 16.1. Dynamics of boolean control networks. 16.2. Controllability. 16.3. Observability. 16.4. Disturbance decoupling. 16.5. Some other control problems -- 17. Game theory. 17.1. An introduction to game theory. 17.2. Infinitely repeated games. 17.3. Local optimization of strategies and local Nash/sub-Nash equilibrium -- 18. Multi-variable polynomials. 18.1. Matrix expression of multi-variable polynomials. 18.2. Differential form of functional matrices. 18.3. Conversion of generators. 18.4. Taylor expansion of multi-variable functions. 18.5. Fundamental formula of differential. 18.6. Lie derivative -- 19. Some applications to differential geometry and algebra. 19.1. Calculation of connection. 19.2. Contraction of tensor field. 19.3. Structure matrix of finite-dimensional algebra. 19.4. Two-dimensional algebras. 19.5. Three-dimensional algebras. 19.6. Lower-dimensional Lie algebra and invertible algebra. 19.7. Tensor product algebra -- 20. Morgan's problem. 20.1. Input-output decomposition. 20.2. Problem formulation. 20.3. Numerical expression of solvability -- 21. Linearization of nonlinear control systems. 21.1. Carleman linearization. 21.2. First integral. 21.3. Invariance of polynomial system. 21.4. Feedback linearization of nonlinear control system. 21.5. Single input feedback linearization. 21.6. Algorithm for non-regular feedback linearization -- 22. Stability region of dynamic systems. 22.1. Stability region. 22.2. Stable submanifold. 22.3. Quadratic approximation. 22.4. Higher order approximation. 22.5. Differential-algebraic system.