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## Book Description

This balanced and comprehensive study presents the theory, methods, and applications of matrix analysis in a new theoretical framework, allowing readers to understand second-order and higher-order matrix analysis in a completely new light. Alongside the core subjects in matrix analysis, such as singular value analysis, solving matrix equations and eigenanalysis, the author introduces new applications and perspectives that are unique to this book. As a very topical subject matter, gradient analysis and optimization plays a central role here. Also included are subspace analysis, projection analysis, and tensor analysis, topics which are often neglected in other books. Having provided a solid foundation to the subject, the author goes on to place particular emphasis on the many applications matrix analysis can have in science and engineering, making this book suitable for scientists, engineers, and graduate students alike.

1. Cover
2. Half-title page
3. Title Page
5. Dedication
6. Contents
7. Preface
8. Notation
9. Abbreviations
10. Algorithms
11. Part I Matrix Algebra
1. 1 Introduction to Matrix Algebra
1. 1.1 Basic Concepts of Vectors and Matrices
2. 1.2 Elementary Row Operations and Applications
3. 1.3 Sets, Vector Subspaces and Linear Mapping
4. 1.4 Inner Products and Vector Norms
5. 1.5 Random Vectors
6. 1.6 Performance Indexes of Matrices
7. 1.7 Inverse Matrices and Pseudo-Inverse Matrices
8. 1.8 Moore–Penrose Inverse Matrices
9. 1.9 Direct Sum and Hadamard Product
10. 1.10 Kronecker Products and Khatri–Rao Product
11. 1.11 Vectorization and Matricization
12. 1.12 Sparse Representations
13. Exercises
2. 2 Special Matrices
1. 2.1 Hermitian Matrices
2. 2.2 Idempotent Matrix
3. 2.3 Permutation Matrix
4. 2.4 Orthogonal Matrix and Unitary Matrix
5. 2.5 Band Matrix and Triangular Matrix
6. 2.6 Summing Vector and Centering Matrix
7. 2.7 Vandermonde Matrix and Fourier Matrix
9. 2.9 Toeplitz Matrix
10. Exercises
3. 3 Matrix Differential
1. 3.1 Jacobian Matrix and Gradient Matrix
2. 3.2 Real Matrix Differential
3. 3.3 Real Hessian Matrix and Identification
5. 3.5 Complex Hessian Matrices and Identification
6. Exercises
12. Part II Matrix Analysis
1. 4 Gradient Analysis and Optimization
2. 4.2 Gradient Analysis of Complex Variable Function
3. 4.3 Convex Sets and Convex Function Identification
4. 4.4 Gradient Methods for Smooth Convex Optimization
5. 4.5 Nesterov Optimal Gradient Method
6. 4.6 Nonsmooth Convex Optimization
7. 4.7 Constrained Convex Optimization
8. 4.8 Newton Methods
9. 4.9 Original–Dual Interior-Point Method
10. Exercises
2. 5 Singular Value Analysis
1. 5.1 Numerical Stability and Condition Number
2. 5.2 Singular Value Decomposition (SVD)
3. 5.3 Product Singular Value Decomposition (PSVD)
4. 5.4 Applications of Singular Value Decomposition
5. 5.5 Generalized Singular Value Decomposition (GSVD)
6. 5.6 Low-Rank–Sparse Matrix Decomposition
7. 5.7 Matrix Completion
8. Exercises
3. 6 Solving Matrix Equations
1. 6.1 Least Squares Method
2. 6.2 Tikhonov Regularization and Gauss–Seidel Method
3. 6.3 Total Least Squares (TLS) Methods
4. 6.4 Constrained Total Least Squares
5. 6.5 Subspace Method for Solving Blind Matrix Equations
6. 6.6 Nonnegative Matrix Factorization: Optimization Theory
7. 6.7 Nonnegative Matrix Factorization: Optimization Algorithms
8. 6.8 Sparse Matrix Equation Solving: Optimization Theory
9. 6.9 Sparse Matrix Equation Solving: Optimization Algorithms
10. Exercises
4. 7 Eigenanalysis
1. 7.1 Eigenvalue Problem and Characteristic Equation
2. 7.2 Eigenvalues and Eigenvectors
3. 7.3 Similarity Reduction
4. 7.4 Polynomial Matrices and Balanced Reduction
5. 7.5 Cayley–Hamilton Theorem with Applications
6. 7.6 Application Examples of Eigenvalue Decomposition
7. 7.7 Generalized Eigenvalue Decomposition (GEVD)
8. 7.8 Rayleigh Quotient
9. 7.9 Generalized Rayleigh Quotient
11. 7.11 Joint Diagonalization
12. Exercises
5. 8 Subspace Analysis and Tracking
1. 8.1 General Theory of Subspaces
2. 8.2 Column Space, Row Space and Null Space
3. 8.3 Subspace Methods
4. 8.4 Grassmann Manifold and Stiefel Manifold
5. 8.5 Projection Approximation Subspace Tracking (PAST)
6. 8.6 Fast Subspace Decomposition
7. Exercises
6. 9 Projection Analysis
1. 9.1 Projection and Orthogonal Projection
2. 9.2 Projectors and Projection Matrices
3. 9.3 Updating of Projection Matrices
4. 9.4 Oblique Projector of Full Column Rank Matrix
5. 9.5 Oblique Projector of Full Row Rank Matrices
6. Exercises
13. Part III Higher-Order Matrix Analysis
1. 10 Tensor Analysis
1. 10.1 Tensors and their Presentation
2. 10.2 Vectorization and Matricization of Tensors
3. 10.3 Basic Algebraic Operations of Tensors
4. 10.4 Tucker Decomposition of Tensors
5. 10.5 Parallel Factor Decomposition of Tensors
6. 10.6 Applications of Low-Rank Tensor Decomposition
7. 10.7 Tensor Eigenvalue Decomposition
8. 10.8 Preprocessing and Postprocessing
9. 10.9 Nonnegative Tensor Decomposition Algorithms
10. 10.10 Tensor Completion
11. 10.11 Software
12. Exercises
14. References
15. Index