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Chapter 1 Linear Space and Mapping 1

1.1 Some Basic Concepts of Abstract Algebra 1

1.1.1 Algebraic Systems 1

1.1.2 Groups 1

1.1.3 Rings 5

1.1.4 Fields 6

1.2 Linear Spaces 7

1.2.1 The Basic Concepts 7

1.2.2 Linear Dependency 9

1.3 Basis of a Linear Space 10

1.3.1 The Notion of a Basis 10

1.3.2 Change of Basis and Transition Matrices 12

1.4 Linear Subspaces 15

1.4.1 The Notion of Linear Subspace 15

1.4.2 Sum and Intersect of Subspaces 16

1.4.3 Direct Sum and Complementary Subspace 20

1.5 Linear Transformations 21

1.5.1 Notion of a Linear Transformation 21

1.5.2 The Matrix Representation of a Linear Transformation 23

1.5.3 Isomorphism on Finite Dimensional Linear Spaces 29

1.5.4 Range and Kernel of a Linear Transformation 30

1.5.5 Composite Transformation 33

1.6 Quotient Space 34

1.6.1 Quotient Space 34

1.6.2 Regular Projection and Induced Transformation 42

1.7 Notes and References 45

1.8 Exercises and Problems 45

Chapter 2 Polynomials and Matrix Polynomials 48

2.1 Linear Algebras 48

2.2 Ring and Euclidean Division 52

2.3 Ideals of Polynomials 56

2.4 Factorization of a Polynomial 60

2.5 Matrix Polynomials 64

2.6 Unimodular λ-Matrix and the Smith Canonical Form 65

2.7 Eleinentary Divisors and Equivalence of Matrix Polynomials 75

2.8 Ideal of Matrix Polynomials and Coprimeness 82

2.9 Notes and References 83

2.10 Problems and Exercises 84

Chapter 3 Linear Transformations 86

3.1 The Eigenvalues of a Linear Transformation 86

3.2 Similarity Reduction，Conditions on Similarity and the Natural Normal Form 93

3.2.1 Conditions on Similarity 93

3.2.2 Similarity Reduction and the Natural Normal Form 95

3.3 The Jordan Canonical Forms in Cn×n and Rn×n 100

3.3.1 The Jordan Canonical Forms in Cn×n 100

3.3.2 The Jordan Canonical Forms in Rn×n 103

3.3.3 The Transition Matrix X 105

3.3.4 Decomposing V into the Direct Sum of Jordan Subspaces 113

3.4 Minimal Polynomials and the First Decomposition of a Linear Space 116

3.4.1 Annihilating and Minimal Polynomials 116

3.4.2 The First Decomposition of a Linear Space 118

3.4.3 Decomposition of a Linear Space V over the Field C 121

3.5 The Cyclic Invariant Subspaces and the Second Decomposition of a Linear Space 125

3.5.1 The Notion of a Cyclic Invariant Subspace 125

3.5.2 The Second Decomposition of a Linear Space 126

3.5.3 Illustrating Examples 129

3.6 Notes and Reference 133

3.7 Problems and Exercises 134

Chapter 4 Linear Transformations in Unitary Spaces 136

4.1 Euclidean and Unitary Spaces 136

4.1.1 The Notions of Euclidean and Unitary Spaces 136

4.1.2 The Characteristics of a Unitary Space 138

4.1.3 The Metric in Unitary Spaces 140

4.2 Orthonormal Basis and the Gram-Schmidt Process 142

4.3 Unitary Transformations 147

4.4 Projectors and Idempotent Matrices 150

4.4.1 Projectors and Idempotent Matrices 150

4.4.2 Orthogonal Complement and Orthogonal Projectors 154

4.5 Adjoint Transformation 156

4.6 Normal Transformations and Normal Matrices 158

4.7 Hermitian Matrices and Hermitian Forms 166

4.7.1 Hermitian Matrices 167

4.7.2 Hermitian Forms 168

4.8 Positive Definite Hermitian Forms 169

4.9 Canonical Forms of a Hermitian Matrix Pair 173

4.10 Rayleigh Quotient 179

4.11 Problems and Exercises 183

Chapter 5 Decomposition of Linear Transformations and Matrices 186

5.1 Spectral Decomposition for Simple Linear Transformations and Matrices 186

5.1.1 Spectral Decomposition of Simple Transformations 186

5.1.2 Spectral Decomposition of Normal Transformations 194

5.2 Singular Value Decomposition for Linear Transformations and Matrices 201

5.3 Full Rank Factorization of Linear Transformations and Matrices 204

5.4 UR and QR Factorizations of Matrices 208

5.5 Polar Factorization ofLinear Transformations and Matrices 210

5.6 Problems and Exercises 214

Chapter 6 Norms for Vectors and Matrices 216

6.1 Norms for Vectors 216

6.2 Norms of Matrices 219

6.3 Induced Norns of Matrices 222

6.4 Sequences of Matrices and the Convergency 227

6.5 Power Series of Matrices 229

6.6 Problems and Exercises 231

Chapter 7 Functions of Matrices 233

7.1 Power Series Representation of a Function of Matrices 233

7.2 Jordan Representation of Functions of Matrices 235

7.3 Polynomial Representation of a Function of Matrices 237

7.4 The Lagrange-Sylvester Interpolation Formula 242

7.5 Exponential and Trigonometric Functions of Matrices 243

7.5.1 Complex Functions of Matrices 243

7.5.2 Real Functions of Matrices 246

7.6 Problems and Exercises 247

Chapter 8 Matrix-valued Functions and Applications to Differential Equations 248

8.1 Matrix-valued Functions 248

8.2 Derivative and Integration ofMatrix-valued Functions 250

8.3 Linear Dependency of Vector-valued Functions 252

8.4 Norms on the Space of Matrix-valued Functions 256

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 Theory 264

8.8 Application of the Matrix Exponential Ⅱ：Controllabilitv and Observability 266

8.8.1 Notion on Controllability 266

8.8.2 Tests for Controllabilitv 268

8.8.3 Observability and the Tests 271

8.8.4 Tests for Observability 272

8.8.5 Essentials ofControllability and Observability 274

8.8.6 State-Feedback and Stabilization 276

8.8.7 Observer Design and Output Injection 278

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

8.8.9 Controllability and Observability Gramian 284

8.8.10 Balanced Realization 286

8.9 Application of the Matrix Exponential Ⅲ：The Hankel Operator 288

8.9.1 The Notion of a Hankel Operator 288

8.9.2 The Singular Values of a Hankel Operator 289

8.9.3 Schmidt Decomposition of a Hankel Operator 290

8.10 Notes and References 293

8.11 Problems and Exercises 293

Chapter 9 Generalized Inverses of Linear Transformations and Matrices 295

9.1 The Generalized Inverse of Linear Transformations and Matrices 295

9.1.1 The Generalized Inverse ofLinear Transformations 295

9.1.2 Generalized Inverses of Matrices 301

9.2 The Reflexive Generalized Inverse of Linear Transformations and Matrices 305

9.2.1 The Reflexive Generalized Inverse of Linear Transformations 305

9.2.2 The Reflexive Generalized Inverse of Matrices 308

9.3 The Pseudo Inverse ofLinear Transformations and Matrices 309

9.4 Generalized Inverse and Applications to Linear Equations 314

9.4.1 Consistent Inhomogeneous Linear Equation 314

9.4.2 Minimum Norm Solution to a Consistent Inhomogeneous Linear Equation 315

9.5 Best Approximation to an Inconsistent Inhomogeneous Linear Equation 317

9.6 Notes and References 319

9.7 Problems and Exercises 319

Chapter 10 Solution to Matrix Equations 320

10.1 The Notion of Kronecker Product and the Properties 320

10.2 Eigenvalues and Eigenvectors of Kronecker Product 324

10.3 Column and Row Expansions of Matrices 326

10.4 Solution to Linear Matrix Equations 327

10.5 Solution to Continuous-time Algebraic Riccati Equations 330

10.6 Solution to Discrete-time Algebraic Riccati Equations 336

10.7 Discussions and Problems 340

Bibliography 343

Notation and Symbols 346

List of Acronyms 348