Solutions to Linear Matrix Equations and Their Applications, livre ebook

This book addresses both the basic and applied aspects of the finite iterative algorithm, CGLS iterative algorithm, and explicit algorithm to some linear matrix equations. The author presents the latest results in three parts: (1) We consider the finite iterative algorithm to the coupled transpose matrix equations and the coupled operator matrix equations with sub-matrix constrained. These two finite iterative algorithms are closely related and progressive. (2) We present MCGLS iterative algorithm for studying least squares problems to the generalized Sylvester-conjugate matrix equation, the generalized Sylvester-conjugate transpose matrix equation, and the coupled linear operator systems, respectively. (3) Compared with the previous two parts, we consider here the explicit solution to some linear matrix equations, which are the nonhomogeneous Yakubovich matrix equation, the nonhomogeneous Yakubovich transpose matrix equation, and the generalized Sylvester matrix equation, respectively. This book is intended for students, researchers, and professionals in the field of numerical algebra, linear matrix equations, nonlinear matrix equations, and control theory.

About the Author ............................................ III

Preface..................................................... V

Notations................................................... VII

CHAPTER 1

Introduction................................................. 1

1.1 The First Part: Finite Iterative Algorithm to Linear Matrix Equation . 1

1.2 The Second Part: MCGLS Iterative Algorithm to Linear Matrix Equation............................................... 3

1.3 The Third Part: Explicit Solutions to Linear Matrix Equation ...... 5

CHAPTER 2

Finite Iterative Algorithm to Coupled Transpose Matrix Equations ....... 11

2.1 Finite Iterative Algorithm and Convergence Analysis ............. 12

2.2 Numerical Example ....................................... 24

2.3 Control Application ...................................... 27

2.4 Conclusions............................................. 28

CHAPTER 3

Finite Iterative Algorithm to Coupled Operator Matrix Equations withSub-Matrix Constrained .................................... 29

3.1Iterative Method for Solving Problem 3.1 ...................... 30

3.2Iterative Method for Solving Problem 3.2 ...................... 41

3.3 Numerical Example ....................................... 42

3.4 Control Application ...................................... 53

3.5 Conclusions............................................. 54

CHAPTER 4

MCGLSIterative Algorithm to Linear Conjugate Matrix Equation ........ 55

4.1 MCGL SIterative Algorithm and Convergence Analysis............ 57

4.2 Numerical Example ....................................... 66

4.3 ControlApplication ...................................... 73

4.4 Conclusions............................................. 74

CHAPTER 5

MCGLS Iterative Algorithm to Linear Conjugate Transpose Matrix Equation................................................... 75

5.1 MCGLS Iterative Algorithm and Convergence Analysis............ 78

5.2 Numerical Example ....................................... 90

5.3 Conclusions............................................. 95

CHAPTER 6

MCGLS Iterative Algorithm to Coupled Linear Operator Systems ........ 97

6.1 Some Useful Lemmas ..................................... 98

6.2 MCGLS Iterative Algorithm and Convergence Analysis............ 99

6.3 Numerical Examples ...................................... 107

6.4 Conclusions............................................. 113

CHAPTER 7

Explicit Solutions to the Matrix Equation X − AXB = CY + R .......... 115

7.1 Solutions to the Real Matrix Equation X − AXB = CY + R ....... 115

7.2 Parametric Pole Assignment for Descriptor Linear Systems by P-DFeedback ........................................ 124

7.3 Conclusions............................................. 128

CHAPTER 8

Explicit Solutions to the Non homogeneous Yakubovich-Transpose Matrix Equation................................................... 131

8.1 The First Approach ...................................... 132

8.2 The Second Approach ..................................... 140

8.3 Illustrative Example ...................................... 142

8.4 Conclusions............................................. 143

CHAPTER 9

Explicit Solutions to the Matrix Equations XB-AX = CY.......................................... 145

9.1 Real Matrix Equation XB − AX = CY and XB - AX = CY  ........................ 145

9.2 Quaternion-j-Conjugate Matrix Equation XB − AX = CY......... 148

9.2.1 Real Representation of a Quaternion Matrix .............. 148

9.2.2 Solutions to the Quaternion j-Conjugate Matrix Equation XB − AX =CY ................................... 150

9.3 Conclusions............................................. 155

CHAPTER 10

Explicit Solutions to Linear Transpose Matrix Equation ................ 157

10.1Solutions to the Sylvester Transpose Matrix Equation ............ 157

10.1.1 The First Case: A or B is Nonsingular ................. 158

10.1.2 The Second Case: A and B are Nonsingular ............. 162

10.2 Solutions to the Generalized Sylvester Transpose Matrix Equation . . 165

10.3 Algorithms for Solving Two Transpose Equations and Numerical Example.............................................. 168

10.4 Application in Control Theory ............................. 174

10.5 Conclusions ............................................ 175

References.................................................. 177