EDP Sciences EDP Sciences EDP Sciences EDP Sciences

Solutions to Linear Matrix Equations and Their Applications

by Caiqin SONG (author)
july 2023
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Presentation

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.

Resume

About theAuthor ............................................ III

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

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

CHAPTER 1

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

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

1.2 TheSecond Part: MCGLS Iterative Algorithm to Linear Matrix

Equation............................................... 3

1.3 TheThird Part: Explicit Solutions to Linear Matrix Equation ...... 5

CHAPTER 2

FiniteIterative Algorithm to Coupled Transpose Matrix Equations ....... 11

2.1 FiniteIterative Algorithm and Convergence Analysis ............. 12

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

2.3 ControlApplication ...................................... 27

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

CHAPTER 3

FiniteIterative 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

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

3.4 ControlApplication ...................................... 53

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

CHAPTER 4

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

4.1 MCGLSIterative Algorithm and Convergence Analysis............ 57

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

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

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

CHAPTER 5

MCGLSIterative Algorithm to Linear Conjugate Transpose Matrix

Equation................................................... 75

5.1 MCGLSIterative Algorithm and Convergence Analysis............ 78

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

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

CHAPTER 6

MCGLSIterative Algorithm to Coupled Linear Operator Systems ........ 97

6.1 SomeUseful Lemmas ..................................... 98

6.2 MCGLSIterative Algorithm and Convergence Analysis............ 99

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

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

CHAPTER 7

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

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

7.2Parametric Pole Assignment for Descriptor Linear Systems

by P-DFeedback ........................................ 124

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

CHAPTER 8

ExplicitSolutions to the Nonhomogeneous Yakubovich-Transpose Matrix

Equation................................................... 131

8.1 TheFirst Approach ...................................... 132

8.2 TheSecond Approach ..................................... 140

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

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

CHAPTER 9

ExplicitSolutions to the Matrix Equations XB



 AXb ¼ CY.......................................... 145

9.1 RealMatrix Equation XB − AX = CY ........................ 145

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

9.2.1 RealRepresentation of a Quaternion Matrix .............. 148

9.2.2Solutions to the Quaternion j-Conjugate Matrix Equation

XB − AXb =CY ................................... 150

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

X Contents

CHAPTER 10

ExplicitSolutions to Linear Transpose Matrix Equation ................ 157

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

10.1.1 TheFirst Case: A or B is Nonsingular ................. 158

10.1.2 TheSecond Case: A and B are Nonsingular ............. 162

10.2 Solutionsto the Generalized Sylvester Transpose Matrix Equation . . 165

10.3Algorithms for Solving Two Transpose Equations and Numerical

Example.............................................. 168

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

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

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

Compléments

Characteristics

Language(s): English

Audience(s): Professionals, Research, Students

Publisher: EDP Sciences & Science Press

Collection: Current Natural Sciences

Published: 20 july 2023

EAN13 (hardcopy): 9782759831029

Reference eBook [PDF]: L31036

EAN13 eBook [PDF]: 9782759831036

Interior: Colour, Black & white

Pages count eBook [PDF]: 196

Size: 2,8 Mo (PDF)

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