
Quaternion Matrix Computations
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Content
- Intro
- QUATERNION MATRIXCOMPUTATIONS
- QUATERNION MATRIXCOMPUTATIONS
- Contents
- Preface
- Acknowledgments
- Notations
- Chapter 1Preliminaries
- 1.1. Introduction
- 1.2. Quaternions
- 1.3. QuaternionMatrices
- 1.4. Eigenvalue Problem
- 1.5. Norms
- 1.5.1. Vector Norms
- 1.5.2. Matrix Norms
- 1.6. Generalized Inverses
- 1.7. Projections
- 1.7.1. Idempotent Matrices and Projections
- 1.7.2. Orthogonal Projections
- 1.7.3. Geometric Meanings of AA┼ and A┼A
- 1.8. Properties of Real RepresentationMatrices
- Chapter 2Computing MatrixDecompositions
- 2.1. ElementaryMatrices
- 1. Gauss Transformation Matrices
- 2. GivensMatrices
- (1) The Real GivensMatrix
- (2) The JRSGivensMatrix
- (3) The qGivensMatrix
- 3. Householder Based Transformations
- (1) The Real Householder Transformation
- (2) Quaternion Householder Based Transformations
- 2.2. The Quaternion LU Decomposition
- 1. The Quaternion LU Decomposition
- 2. The Partial Pivoting Quaternion LU Decomposition
- 2.3. The Quaternion LDLH and CholeskyDecompositions
- 2.4. The Quaternion QR Decomposition
- 2.4.1. The Quaternion Householder QRD
- 2.4.2. The Givens QRD
- 2.4.3. The Modified Gram-Schimit Scheme
- 2.4.4. Complete Orthogonal Decomposition
- 2.5. The Quaternion SVD
- Chapter 3Linear System and GeneralizedLeast Squares Problems
- 3.1. Linear System
- 1. Homogeneous Linear System
- 2. Nonhomogeneous Linear System
- 3.2. The Linear Least Squares Problem
- 3.2.1. The LS Problem and Its Equivalent Problems
- 1. The Normal Equation
- 2. The KKT Equation
- 3.2.2. The Regularization of the LS Problem
- 1. The Truncated LS Problem
- 2. The Tikhonov Regularization
- 3.2.3. Some Matrix Equations
- 3.3. The Total Least Squares Problem
- 3.4. The Equality Constrained Least Squares Problem
- 1. The KKT Equation
- 2. The Unitary Decomposition Method
- 3. The Q-SVD Method
- 4. The Weighted LS Method
- 5. Unconstrained LS Method
- Chapter 4Direct Methods for SolvingLinear System and GeneralizedLS Problems
- 4.1. DirectMethods for Linear System
- 4.2. DirectMethods for the LS Problem
- 1. The QR Decomposition Method
- 2. The Normal EquationMethod
- 3. The Complete Orthogonal Decomposition Method
- 4. The SVD Method
- 4.3. DirectMethods for the TLS Problem
- 1. Basic SVD Method
- 2. The Complete Orthogonal Decomposition Method
- 3. The Cholesky DecompositionMethod
- 4.4. DirectMethods for the LSE Problem
- 1. Null Space Method
- 2. The Weighted LS Method
- 3. Direct EliminationMethod
- 4. The QR Decomposition and the Q-SVDMethods
- 4.5. Some Matrix Equations
- 1. Special Solutions to the QuaternionMatrix Equation AX = B
- 2. Special Solutions to the QuaternionMatrix EquationAXB+CXD = E
- Chapter 5Iterative Methods for SolvingLinear System and GeneralizedLS Problems
- 5.1. Basic Knowledge
- 5.1.1. The Chebyshev Polynomials
- 5.1.2. The Range of Eigenvalues of Real Symmetric TridiagonalMatrices
- 5.2. Iterative Methods for Linear System
- 5.2.1. Basic Theory of Splitting IterativeMethod
- 1. The Jacobi Iteration
- 2. The Gauss-Seidel Iteration
- 3. The Successive over Relaxation Iteration (SOR)
- 4. The Chebyshev Semi-IterativeMethod
- 5.2.2. The Krylov Subspace Methods
- 1. The Conjugate GradientMethod (CG)
- 5.3. Iterative Methods for the LS Problem
- 5.3.1. Splitting IterativeMethods
- 1. The Jacobi Iteration
- 2. The Gauss-Seidel Iteration
- 4. The Chebyshev Semi-Iterative Acceleration
- 3. The Successive over Rrelaxation Iteration (SOR)
- 5.3.2. The Krylov Subspace Methods
- 1. The Conjugate GradientMethod (CGLS)
- 2. The QR Least SquaresMethod (LSQR)
- 5.3.3. Preconditioning Hermitian-Skew Hermitian Splitting Itera-tionMethods
- 5.4. Iterative Methods for the TLS Problem
- 5.4.1. The Partial SVD Method
- 5.4.2. BidiagonalizationMethod
- 5.5. Some Matrix Equations
- Chapter 6Computations of QuaternionEigenvalue Problems
- 6.1. Quaternion Hermitian Right Eigenvalue Problem
- 6.1.1. The Power Method and Inverse Power Method forQuaternion Hermitian Right Eigenvalue Problem
- 1. The Power Method for Quaternion Hermitian Right EigenvalueProblem
- 2. The Inverse Power Method for Quaternion Hermitian RightEigenvalue Problem
- 6.1.2. Real Structure-Preserving Algorithm of Hermitian QRAlgorithm for Hermitian Right Eigenvalue Problem
- 6.1.3. Real Structure-Preserving Algorithm of the Jacobi Methodfor Hermitian Right Eigenvalue Problem
- 6.1.4. Subspace Methods
- 1. The Rayleigh-Ritz Projection Method
- 2. The Hermitian Lanczos Method
- 6.2. Quaternion Non-Hermitian Right EigenvalueProblem
- 6.2.1. The Power Method and the Inverse Power Method
- 1. The Power Method for Quaternion Non-Hermitian RightEigenvalue Problem
- 2. The Inverse Power Method for Quaternion Non-Hermitian RightEigenvalue Problem
- 6.2.2. The Quaternion QR Algorithm for QuaternionNon-Hermitian Right Eigenvalue Problem
- 3. Implicit Double Shift Trick
- References
- About the Authors
- Index
- Blank Page
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