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Contents Chapter 1 Introduction 1 Chapter 2 Basic Quaternion Matrix Theory 7 2.1 Quaternion Matrices 7 2.2 Quaternion Matrix Eigenvalue Problems 8 2.3 Unitary Quaternion Transformations 10 2.3.1 Improved Householder-Based Transformations 10 2.3.2 Generalized Quaternion Givens Transformations 12 2.4 Complex and Real Counterpart Methods 13 2.5 JRS-Symmetric Matrices 18 Chapter 3 General Quaternion Matrix Eigenvalue Problem 22 3.1 Structure-Preserving QR Algorithm 22 3.1.1 The Upper JRS-Hessenberg Form 23 3.1.2 Structure-Preserving Decompositions 25 3.1.3 The Structure-Preserving JRS-Hessenberg QR Iteration 27 3.2 Quaternion QR Algorithm 35 3.2.1 The Quaternion Hessenberg Reduction 36 3.2.2 Quaternion Hessenberg QR Factorization 39 3.2.3 Implicit Double Shift Quaternion QR Algorithm 40 3.2.4 Numerical Examples 44 3.3 The Power and Inverse Power Methods 48 3.3.1 The Power Method 48 3.3.2 The Inverse Power Method 58 3.4 Perturbation Theory 60 3.4.1 The Perturbation of Eigenvalues 61 3.4.2 Simple Eigenpairs 65 3.5 Conclusion 68 Chapter 4 Hermitian Quaternion Matrix Eigenvalue Problem 69 4.1 Background 69 4.2 2¡Á2 Block Structure Preserving Method 71 4.2.1 Structure-Preserving Method 71 4.2.2 Structure-Preserving Algorithm 804.2.3 Numerical Examples 82 4.3 4¡Á4 Block Structure-Preserving Method 86 4.3.1 Structure-Preserving Tridiagonalizing 87 4.3.2 Right Eigenvalue Problem 92 4.3.3 Structure-Preserving Algorithm 92 4.3.4 Numerical Examples 96 4.4 Structure-Preserving Jacobi Algorithm 100 4.4.1 History of Jacobi Algorithm 100 4.4.2 Structure-Preserving Jacobi Algorithm 101 4.4.3 Numerical Examples 111 4.5 Lanczos Method for Large-Scale Quaternion Singular Value Decomposition 114 4.5.1 History of Lancozos Method 115 4.5.2 The Quaternion Lanczos Method 115 4.5.3 Lanczos-Based Algorithms 118 4.5.4 Numerical Examples 122 4.6 Conclusion 125 Chapter 5 Applications 127 5.1 Quaternion Principal Component Analysis 127 5.1.1 Representation and Compression of Color Face Images 129 5.1.2 Face Recognition in Color 129 5.1.3 Experiments 131 5.2 Two Dimensional Quaternion Principal Component Analysis 135 5.2.1 Color 2DPCA Approach 135 5.2.2 Experiments 138 5.3 Color Image Inpainting 141 5.3.1 Preliminaries 141 5.3.2 Robust Quaternion Matrix Completion 142 5.3.3 Experiments 144 5.4 Color Watermarking 149 5.4.1 Embedding and Extracting Procedure 149 5.4.2 Evaluation Criteria 152 5.4.3 Experiments 152 5.5 Conclusion 154 Bibliography 156 Book list of the Series in Information and Computational Science 162 Color Figures