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January 20, 2011 14:43 World Scientific Book - 9in x 6in book Contents Preface v Symbol Index xiii 1. Matrix Calculus 1 1.1 Definitions and Notation . . . . . . . . . . . . . . . . . . . 1 1.2 Matrix Operations . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Linear Equations . . . . . . . . . . . . . . . . . . . . . . . 10 1.4 Trace and Determinant . . . . . . . . . . . . . . . . . . . 11 1.5 Eigenvalue Problem . . . . . . . . . . . . . . . . . . . . . 18 1.6 Cayley-Hamilton Theorem . . . . . . . . . . . . . . . . . . 24 1.7 Projection Matrices . . . . . . . . . . . . . . . . . . . . . 26 1.8 Fourier and Hadamard Matrices . . . . . . . . . . . . . . 29 1.9 Transformation of Matrices . . . . . . . . . . . . . . . . . 35 1.10 Permutation Matrices . . . . . . . . . . . . . . . . . . . . 39 1.11 Matrix Decompositions . . . . . . . . . . . . . . . . . . . 42 1.12 Pseudo Inverse . . . . . . . . . . . . . . . . . . . . . . . . 50 1.13 Vec Operator . . . . . . . . . . . . . . . . . . . . . . . . . 52 1.14 Vector and Matrix Norms . . . . . . . . . . . . . . . . . . 54 1.15 Rank-k Approximations . . . . . . . . . . . . . . . . . . . 60 1.16 Sequences of Vectors and Matrices . . . . . . . . . . . . . 61 1.17 Gram-Schmidt Orthonormalization . . . . . . . . . . . . . 65 1.18 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 1.19 Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 75 1.20 Commutators and Anti-Commutators . . . . . . . . . . . 78 1.21 Functions of Matrices . . . . . . . . . . . . . . . . . . . . 79 ix MATRIX CALCULUS AND KRONECKER PRODUCT - A Practical Approach to Linear and Multilinear Algebra (Second Edition) © World Scientific Publishing Co. Pte. Ltd. http://www.worldscibooks.com/mathematics/8030.html January 20, 2011 14:43 World Scientific Book - 9in x 6in book x Matrix Calculus and Kronecker Product 2. Kronecker Product 83 2.1 Definitions and Notations . . . . . . . . . . . . . . . . . . 83 2.2 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . 90 2.3 Matrix Multiplication . . . . . . . . . . . . . . . . . . . . 94 2.4 Permutation Matrices . . . . . . . . . . . . . . . . . . . . 100 2.5 Trace and Determinant . . . . . . . . . . . . . . . . . . . 104 2.6 Eigenvalue Problem . . . . . . . . . . . . . . . . . . . . . 108 2.7 Projection Matrices . . . . . . . . . . . . . . . . . . . . . 112 2.8 Fourier and Hadamard Matrices . . . . . . . . . . . . . . 114 2.9 Direct Sum . . . . . . . . . . . . . . . . . . . . . . . . . . 118 2.10 Kronecker Sum . . . . . . . . . . . . . . . . . . . . . . . . 121 2.11 Matrix Decompositions . . . . . . . . . . . . . . . . . . . 123 2.12 Vec Operator . . . . . . . . . . . . . . . . . . . . . . . . . 130 2.13 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 2.14 Group Representation Theory . . . . . . . . . . . . . . . . 137 2.15 Commutators and Anti-Commutators . . . . . . . . . . . 142 2.16 Inversion of Partitioned Matrices . . . . . . . . . . . . . . 144 2.17 Nearest Kronecker Product . . . . . . . . . . . . . . . . . 146 3. Applications 153 3.1 Spin Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 153 3.2 Pauli Group, Clifford Groups and Bell Group . . . . . . . 161 3.3 Application in Quantum Theory . . . . . . . . . . . . . . 163 3.4 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . 170 3.5 One Dimensional Ising Model . . . . . . . . . . . . . . . . 172 3.6 Fermi Systems . . . . . . . . . . . . . . . . . . . . . . . . 177 3.7 Dimer Problem . . . . . . . . . . . . . . . . . . . . . . . . 188 3.8 Two Dimensional Ising Model . . . . . . . . . . . . . . . . 196 3.9 One Dimensional Heisenberg Model . . . . . . . . . . . . 215 3.10 Hopf Algebras . . . . . . . . . . . . . . . . . . . . . . . . . 226 3.11 Quantum Groups . . . . . . . . . . . . . . . . . . . . . . . 227 3.12 Lax Representation . . . . . . . . . . . . . . . . . . . . . . 231 3.13 Signal Processing . . . . . . . . . . . . . . . . . . . . . . . 234 3.14 Clebsch-Gordan Series . . . . . . . . . . . . . . . . . . . . 242 3.15 Braid-like Relations . . . . . . . . . . . . . . . . . . . . . 245 3.16 Fast Fourier Transform . . . . . . . . . . . . . . . . . . . . 249 3.17 Image Compression . . . . . . . . . . . . . . . . . . . . . . 253 MATRIX CALCULUS AND KRONECKER PRODUCT - A Practical Approach to Linear and Multilinear Algebra (Second Edition) © World Scientific Publishing Co. Pte. Ltd. http://www.worldscibooks.com/mathematics/8030.html January 20, 2011 14:43 World Scientific Book - 9in x 6in book Contents xi 4. Tensor Product 255 4.1 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . 255 4.2 Hilbert Tensor Products of Hilbert Spaces . . . . . . . . . 263 4.3 Spin and Statistics for the n-Body Problem . . . . . . . . 268 4.4 Exciton-Phonon Systems . . . . . . . . . . . . . . . . . . . 271 4.5 Interpretation of Quantum Mechanics . . . . . . . . . . . 274 4.6 Universal Enveloping Algebra . . . . . . . . . . . . . . . . 280 5. Computer Algebra Implementations 283 Bibliography 301 Index 305 MATRIX CALCULUS AND KRONECKER PRODUCT - A Practical Approach to Linear and Multilinear Algebra (Second Edition) © World Scientific Publishing Co. Pte. Ltd. http://www.worldscibooks.com/mathematics/8030.html
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