The basic contents of matrix wheel include: linear space and linear operator, inner product space and equal product transformation, λ moment Chen and Jordan canonical form, normed linear space and matrix norm, matrix calculus operation and its application, generalized inverse matrix and its application, matrix decomposition, matrix Kronecker product, Hadamard product and inverse product, and several special matrices (such as non-negative matrix and positive matrix, cyclic matrix and prime matrix, random matrix and double random matrix, etc.).

The following is the catalogue of Matrix Theory published by Tsinghua University Publishing House on 20 13:

first part

Chapter 65438 +0 linear operator 3 on linear space

1. 1 linear space 3

1. 1. 1 definition and basic properties of linear space 3

1. 1.2 foundation, dimensions and coordinates 8

* 1. 1.3 linear subspace 15

Exercise 1. 12 1

1.2 linear operator and its matrix 24

1.2. 1 linear operator on linear space 24

1.2.2 isomorphism operator and linear space isomorphism 27

Matrix representation of 1.2.3 linear operator 29

1.2.4 operation of linear operator 3 1

1.2.5 linear transformation and square matrix 34

1.2.6 eigenvalue problem of linear transformation 42

* 1.2.7 Invariant subspace of linear transformation 54

Exercise 1.256

Chapter 2 Equal product transformation on inner product space 62

2. 1 internal product space 62

2. 1. 1 inner product and Euclidean space 63

2. 1.2 Introduction to Unitary Space 73

Exercise 2. 174

2.2 Equal product transformation and its matrix 77

2.2. 1 orthogonal transformation and orthogonal matrix 78

2.2.2 Two Commonly Used Orthogonal Transforms and Their Matrices 85

*2.2.3 Introduction to Unitary Transformation and Unitary Matrix 95

*2.2.4 Orthogonal projection transformation and orthogonal projection matrix 96

Exercise 2.2 10 1

*2.3 Hermite transform and its matrix 103

2.3. 1 Symmetric transformation and Hermite transformation 103

2.3.2 Hermite positive and semi-positive definite matrices 106

2.3.3 matrix inequality 109

2.3.4 Properties of Hermite Matrix Eigenvalues 1 1 1

2.3.5 General complex positive definite matrix 1 14

2.3.6 Normal matrix 1 15

Exercise 2.3 1 17

Chapter 3 λ matrix and Jordan canonical form 1 19.

3. 1λ matrix 1 19

3. The concept of1.1λ matrix 1 19

3. The normal form of1.2λ matrix at offset 122

3. 1.3 Invariant factor and elementary factor 124

3.2 Jordan Standard Form 136

3.2. The number1is matrixing into a similar Jordan canonical form 136.

3.2.2 Application of Jordan Standard Form 147

3.3 Gloria? Hamilton Theorem and Minimum Polynomial 149

Exercise 3 155

Chapter 4 Normal Linear Spaces and Matrix Norms 158

4. 1 normalized linear space 158

Norm 4. 1. 1 vector 158

4. Properties of1.2 Vector Norm 165

Exercise 4. 1 167

4.2 norm of matrix 168

4.2. Definition and properties of1matrix norm 168

4.2.2 Operator Specification 170

4.2.3 Properties of spectral norm and spectral radius 176

Exercise 4.2 179

4.3 perturbation analysis and condition number of matrix 180

4.3. 1 ill-conditioned equation and ill-conditioned matrix 18 1

4.3.2 Conditional number of matrix 18 1

*4.3.3 Perturbation Analysis of Matrix Eigenvalue 185

Exercise 4.3 189

Chapter 5 Matrix Analysis and Its Application 192

5. 1 limit of vector sequence and matrix sequence 192

Limit 5. 1. 1 vector sequence 192

5. 1.2 matrix sequence limit 194

5.2 matrix series and matrix function 198

5.2. 1 matrix series 198

Matrix function 206

5.3 Differential and Integral of Function Matrix 2 16

5.3. 1 derivative of function matrix to real variable 2 17

5.3.2 Special derivative of function matrix 22 1

5.3.3 Total Differential of Matrix 226

5.3.4 Integration of Function Matrix 228

*5.4 matrix differential equation 229

5.4. 1 Solutions of homogeneous linear differential equations with constant coefficients 229

5.4.2 Solution of Non-homogeneous Linear Differential Equation with Constant Coefficient236

5.4.3n Solutions of n-order differential equations with constant coefficients 239

Exercise 5244

The second part of a two-part program/book.

Chapter VI Generalized Inverse Matrix and Its Application 25 1

6. Some generalized inverses of1matrix

6. 1. 1 Basic concept of generalized inverse matrix

6. 1.2 minus anti-A-252

6. 1.3 reflexive minus sign against A-r256

6. 1.4 Minimum norm generalized inverse A-m262

6. 1.5 Least Squares Generalized Inverse A-l265

6. 1.6 plus anti-A+267

6.2 the application of generalized inverse in solving linear equations 273

Formula 274 for solving linear equations

6.2.2 Compatibility Equation and General Solution of A-274

6.2.3 Compatibility Equation and Minimum Norm Solution of A-m277

6.2.4 Least Square Solution of Contradictory Equation and A-l28 1

6.2.5 Least Square Solution of Linear Equations and A+286

Exercise 6288

Chapter 7 Matrix Decomposition 29 1

7. Triangular decomposition of1matrix x29 1

7. 1. 1 Matrix Description of Elimination Process 29 1

7. Triangular Decomposition of1.2 Matrix 295

7. 1.3 Common Trigonometric Decomposition Formula 300

7.2 QR (Orthogonal Triangle) Decomposition of Matrix 306

7.2. 1QR decomposition concept 306

7.2.2QR Practical solution of QR decomposition 309

7.3 Maximum Rank Decomposition of Matrix 3 16

7.4 Singular Value Decomposition and Spectral Decomposition 320

7.4. 1 matrix singular value decomposition 320

7.4.2 Spectral Decomposition of Simple Matrices 324

Exercise 7326

Chapter VIII Several Special Matrices 330

8. 1 nonnegative matrix 330

8. 1. 1 nonnegative matrix and positive matrix 330

8. 1.2 Irreducible Nonnegative Matrix 336

8. 1.3 Prime Matrix and Cyclic Matrix 342

8.2 Random Matrix and Double Random Matrix 343

8.3 Monotone Matrix 346

8.4M matrix and H matrix 348

8.4. 1M matrix 348

8.4.2H matrix 353

8.5T matrix and Hankel matrix 354

Exercise 8357

Chapter 9 Special product of matrix and its application 358

9. 1 Kroneck product 358

9. The concept of1.1Kronecker product 358

9. 1.2 Properties of Kronecker product 359

9.2 Adama products 364

9.3 Hadamard product of inverse product and nonnegative matrix 366

9.4 Kroneck product application example 366

9.4. 1 matrix straightening 367

9.4.2 Solution of Linear Matrix Equation 368

Exercise 9370

Chapter 10 Introduction to Symplectic Space and Symplectic Transformation 37 1

10. 1 antisymmetric bilinear function and symplectic space 372

10. 1. 1 antisymmetric bilinear function 372

Outer product of 10. 1.2 linear function 372

10. 1.3 The definition of symplectic space 373

Antisymmetric orthogonal complement of 10.2 subspace 374

10.2. 1 antisymmetric orthogonal complement 374

10.2.2 Several Special Subspaces 378

Properties of 10.2.3 symplectic space

10.2.4 octyl 379

10.3 symplectic transformation and symplectic matrix 380

10.3. 1 symplectic transformation and its matrix 380

Eigenvalues of 10.3.2 symplectic transformation 383

10.4 symplectic involution 385

Exercise 10390