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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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