自动控制中的线性代数 英文【2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载】

自动控制中的线性代数 英文PDF格式电子书版下载

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Chapter 1 Linear Space and Mapping1

1.1 Some Basic Concepts of Abstract Algebra1

1.1.1 Algebraic Systems1

1.1.2 Groups1

1.1.3 Rings5

1.1.4 Fields6

1.2 Linear Spaces7

1.2.1 The Basic Concepts7

1.2.2 Linear Dependency9

1.3 Basis of a Linear Space10

1.3.1 The Notion of a Basis10

1.3.2 Change of Basis and Transition Matrices12

1.4 Linear Subspaces15

1.4.1 The Notion of Linear Subspace15

1.4.2 Sum and Intersect of Subspaces16

1.4.3 Direct Sum and Complementary Subspace20

1.5 Linear Transformations21

1.5.1 Notion of a Linear Transformation21

1.5.2 The Matrix Representation of a Linear Transformation23

1.5.3 Isomorphism on Finite Dimensional Linear Spaces29

1.5.4 Range and Kernel of a Linear Transformation30

1.5.5 Composite Transformation33

1.6 Quotient Space34

1.6.1 Quotient Space34

1.6.2 Regular Projection and Induced Transformation42

1.7 Notes and References45

1.8 Exercises and Problems45

Chapter 2 Polynomials and Matrix Polynomials48

2.1 Linear Algebras48

2.2 Ring and Euclidean Division52

2.3 Ideals of Polynomials56

2.4 Factorization of a Polynomial60

2.5 Matrix Polynomials64

2.6 Unimodular λ-Matrix and the Smith Canonical Form65

2.7 Eleinentary Divisors and Equivalence of Matrix Polynomials75

2.8 Ideal of Matrix Polynomials and Coprimeness82

2.9 Notes and References83

2.10 Problems and Exercises84

Chapter 3 Linear Transformations86

3.1 The Eigenvalues of a Linear Transformation86

3.2 Similarity Reduction，Conditions on Similarity and the Natural Normal Form93

3.2.1 Conditions on Similarity93

3.2.2 Similarity Reduction and the Natural Normal Form95

3.3 The Jordan Canonical Forms in Cn×n and Rn×n100

3.3.1 The Jordan Canonical Forms in Cn×n100

3.3.2 The Jordan Canonical Forms in Rn×n103

3.3.3 The Transition Matrix X105

3.3.4 Decomposing V into the Direct Sum of Jordan Subspaces113

3.4 Minimal Polynomials and the First Decomposition of a Linear Space116

3.4.1 Annihilating and Minimal Polynomials116

3.4.2 The First Decomposition of a Linear Space118

3.4.3 Decomposition of a Linear Space V over the Field C121

3.5 The Cyclic Invariant Subspaces and the Second Decomposition of a Linear Space125

3.5.1 The Notion of a Cyclic Invariant Subspace125

3.5.2 The Second Decomposition of a Linear Space126

3.5.3 Illustrating Examples129

3.6 Notes and Reference133

3.7 Problems and Exercises134

Chapter 4 Linear Transformations in Unitary Spaces136

4.1 Euclidean and Unitary Spaces136

4.1.1 The Notions of Euclidean and Unitary Spaces136

4.1.2 The Characteristics of a Unitary Space138

4.1.3 The Metric in Unitary Spaces140

4.2 Orthonormal Basis and the Gram-Schmidt Process142

4.3 Unitary Transformations147

4.4 Projectors and Idempotent Matrices150

4.4.1 Projectors and Idempotent Matrices150

4.4.2 Orthogonal Complement and Orthogonal Projectors154

4.5 Adjoint Transformation156

4.6 Normal Transformations and Normal Matrices158

4.7 Hermitian Matrices and Hermitian Forms166

4.7.1 Hermitian Matrices167

4.7.2 Hermitian Forms168

4.8 Positive Definite Hermitian Forms169

4.9 Canonical Forms of a Hermitian Matrix Pair173

4.10 Rayleigh Quotient179

4.11 Problems and Exercises183

Chapter 5 Decomposition of Linear Transformations and Matrices186

5.1 Spectral Decomposition for Simple Linear Transformations and Matrices186

5.1.1 Spectral Decomposition of Simple Transformations186

5.1.2 Spectral Decomposition of Normal Transformations194

5.2 Singular Value Decomposition for Linear Transformations and Matrices201

5.3 Full Rank Factorization of Linear Transformations and Matrices204

5.4 UR and QR Factorizations of Matrices208

5.5 Polar Factorization ofLinear Transformations and Matrices210

5.6 Problems and Exercises214

Chapter 6 Norms for Vectors and Matrices216

6.1 Norms for Vectors216

6.2 Norms of Matrices219

6.3 Induced Norns of Matrices222

6.4 Sequences of Matrices and the Convergency227

6.5 Power Series of Matrices229

6.6 Problems and Exercises231

Chapter 7 Functions of Matrices233

7.1 Power Series Representation of a Function of Matrices233

7.2 Jordan Representation of Functions of Matrices235

7.3 Polynomial Representation of a Function of Matrices237

7.4 The Lagrange-Sylvester Interpolation Formula242

7.5 Exponential and Trigonometric Functions of Matrices243

7.5.1 Complex Functions of Matrices243

7.5.2 Real Functions of Matrices246

7.6 Problems and Exercises247

Chapter 8 Matrix-valued Functions and Applications to Differential Equations248

8.1 Matrix-valued Functions248

8.2 Derivative and Integration ofMatrix-valued Functions250

8.3 Linear Dependency of Vector-valued Functions252

8.4 Norms on the Space of Matrix-valued Functions256

8.5 The Differential Equation ？（t）＝A（t）X（t）259

8.6 Solution to the State Equation ？（t）＝Ax（t）＋Bu（t）263

8.7 Application of the Matrix Exponential Ⅰ：The Stability Theory264

8.8 Application of the Matrix Exponential Ⅱ：Controllabilitv and Observability266

8.8.1 Notion on Controllability266

8.8.2 Tests for Controllabilitv268

8.8.3 Observability and the Tests271

8.8.4 Tests for Observability272

8.8.5 Essentials ofControllability and Observability274

8.8.6 State-Feedback and Stabilization276

8.8.7 Observer Design and Output Injection278

8.8.8 Co-prime Factorization of a Transfer Function Matrix over H∞280

8.8.9 Controllability and Observability Gramian284

8.8.10 Balanced Realization286

8.9 Application of the Matrix Exponential Ⅲ：The Hankel Operator288

8.9.1 The Notion of a Hankel Operator288

8.9.2 The Singular Values of a Hankel Operator289

8.9.3 Schmidt Decomposition of a Hankel Operator290

8.10 Notes and References293

8.11 Problems and Exercises293

Chapter 9 Generalized Inverses of Linear Transformations and Matrices295

9.1 The Generalized Inverse of Linear Transformations and Matrices295

9.1.1 The Generalized Inverse ofLinear Transformations295

9.1.2 Generalized Inverses of Matrices301

9.2 The Reflexive Generalized Inverse of Linear Transformations and Matrices305

9.2.1 The Reflexive Generalized Inverse of Linear Transformations305

9.2.2 The Reflexive Generalized Inverse of Matrices308

9.3 The Pseudo Inverse ofLinear Transformations and Matrices309

9.4 Generalized Inverse and Applications to Linear Equations314

9.4.1 Consistent Inhomogeneous Linear Equation314

9.4.2 Minimum Norm Solution to a Consistent Inhomogeneous Linear Equation315

9.5 Best Approximation to an Inconsistent Inhomogeneous Linear Equation317

9.6 Notes and References319

9.7 Problems and Exercises319

Chapter 10 Solution to Matrix Equations320

10.1 The Notion of Kronecker Product and the Properties320

10.2 Eigenvalues and Eigenvectors of Kronecker Product324

10.3 Column and Row Expansions of Matrices326

10.4 Solution to Linear Matrix Equations327

10.5 Solution to Continuous-time Algebraic Riccati Equations330

10.6 Solution to Discrete-time Algebraic Riccati Equations336

10.7 Discussions and Problems340

Bibliography343

Notation and Symbols346

List of Acronyms348