內容大鋼
This book is intended to provide the fundamental material for young researchers of the quaternion matrix eigenvalue problem. Starting from the origin of the right eigenvalue problem of quaternion matrices, we introduce the basic theory and methods of quaternion matrices in the first chapter. In the second chapter, we study the eigenvalue problem of general quaternion matrices, including the structure-preserving QR algorithm, the quaternion QR algorithms, etc. In the third chapter, we research the eigenvahie problem of Hermitian quaternion matrices, with proposing two kinds of direct methods using the Householder transform and two iterative methods: Jacobi algorithm and Lanczos method. In the last chapter, we provide several practical application models, such as two dimensional principle component analysis and color image inpainting, which relying on solving the quaternion eigenvalue problem. The writing of this book is straight forward and is addressed to readers who have a basic graduate mathematics background. Alternatively, it may be used in a beginning graduate level course and as a reference.
目錄
Chapter 1 Introduction
Chapter 2 Basic Quaternion Matrix Theory
2.1 Quaternion Matrices
2.2 Quaternion Matrix Eigenvalue Problems
2.3 Unitary Quaternion Transformations
2.3.1 Improved Householder-Based Transformations
2.3.2 Generalized Quaternion Givens Transformations
2.4 Complex and Real Counterpart Methods
2.5 JRS-Symmetric Matrices
Chapter 3 General Quaternion Matrix Eigenvalue Problem
3.1 Structure-Preserving QR Algorithm
3.1.1 The Upper JRS-Hessenberg Form
3.1.2 Structure-Preserving Decompositions
3.1.3 The Structure-Preserving JRS-Hessenberg QR Iteration
3.2 Quaternion QR Algorithm
3.2.1 The Quaternion Hessenberg Reduction
3.2.2 Quaternion Hessenberg QR Factorization
3.2.3 Implicit Double Shift Quaternion QR Algorithm
3.2.4 Numerical Examples
3.3 The Power and Inverse Power Methods
3.3.1 The Power Method
3.3.2 The Inverse Power Method
3.4 Perturbation Theory
3.4.1 The Perturbation of Eigenvalues
3.4.2 Simple Eigenpairs
3.5 Conclusion
Chapter 4 Hermitian Quaternion Matrix Eigenvalue Problem
4.1 Background
4.2 2×2 Block Structure Preserving Method
4.2.1 Structure-Preserving Method
4.2.2 Structure-Preserving Algorithm
4.2.3 Numerical Examples
4.3 4×4 Block Structure-Preserving Method
4.3.1 Structure-Preserving Tridiagonalizing
4.3.2 Right Eigenvalue Problem
4.3.3 Structure-Preserving Algorithm
4.3.4 Numerical Examples
4.4 Structure-Preserving Jacobi Algorithm
4.4.1 History of Jacobi Algorithm
4.4.2 Structure-Preserving Jacobi Algorithm
4.4.3 Numerical Examples
4.5 Lanczos Method for Large-Scale Quaternion Singular Value Decomposition
4.5.1 History of Lancozos Method
4.5.2 The Quaternion Lanczos Method
4.5.3 Lanczos-Based Algorithms
4.5.4 Numerical Examples
4.6 Conclusion
Chapter 5 Applications
5.1 Quaternion Principal Component Analysis
5.1.1 Representation and Compression of Color Face Images
5.1.2 Face Recognition in Color
5.1.3 Experiments
5.2 Two Dimensional Quaternion Principal Component Analysis
5.2.1 Color 2DPCA Approach
5.2.2 Experiments
5.3 Color Image Inpainting
5.3.1 Preliminaries
5.3.2 Robust Quaternion Matrix Completion
5.3.3 Experiments
5.4 Color Watermarking
5.4.1 Embedding and Extracting Procedure
5.4.2 Evaluation Criteria
5.4.3 Experiments
5.5 Conclusion
Bibliography
Book list of the Series in Information and Computational Science
Color Figures
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