Preface ........................................................ xi
Note to the Reader ............................................. xv
Chapter 1. Preliminaries ........................................ 1
1.1 Vector Spaces .............................................. 1
1.2 Bases and Coordinates ...................................... 3
1.3 Linear Transformations ..................................... 3
1.4 Matrices ................................................... 4
1.5 The Matrix of a Linear Transformation ...................... 5
1.6 Change of Basis and Similarity ............................. 6
1.7 Transposes ................................................. 8
1.8 Special Types of Matrices .................................. 8
1.9 Submatrices, Partitioned Matrices, and Block
Multiplication ............................................. 9
1.10 Invariant Subspaces ....................................... 10
1.11 Determinants .............................................. 11
1.12 Tensor Products ........................................... 13
Exercises ...................................................... 14
Chapter 2. Inner Product Spaces and Orthogonality .............. 17
2.1 The Inner Product ......................................... 17
2.2 Length, Orthogonality, and Projection onto a Line ......... 18
2.3 Inner Products in  n ...................................... 21
2.4 Orthogonal Complements and Projection onto a Subspace ..... 23
2.5 Hilbert Spaces and Fourier Series ......................... 27
2.6 Unitary Tranformations .................................... 31
2.1 The Gram-Schmidt Process and QR Factorization ............. 33
2.8 Linear Functional and the Dual Space ...................... 35
Exercises ...................................................... 36
Chapter 3. Eigenvalues, Eigenvectors, Diagonalization, and
Triangularization .............................................. 39
3.1 Eigenvalues ............................................... 39
3.2 Algebraic and Geometric Multiplicity ...................... 40
3.3 Diagonalizability ......................................... 41
3.4 A Triangularization Theorem ............................... 44
3.5 The Gerњgorin Circle Theorem .............................. 45
3.6 More about the Characteristic Polynomial .................. 46
3.7 Eigenvalues of AB and ВA .................................. 48
Exercises ...................................................... 48
Chapter 4. The Jordan and Weyr Canonical Forms ................. 51
4.1 A Theorem of Sylvester and Reduction to Block Diagonal
Form ...................................................... 53
4.2 Nilpotent Matrices ........................................ 57
4.3 The Jordan Form of a General Matrix ....................... 63
4.4 The Cayley-Hamilton Theorem and the Minimal Polynomial .... 64
4.5 Weyr Normal Form .......................................... 67
Exercises ...................................................... 74
Chapter 5. Unitary Similarity and Normal Matrices .............. 77
5.1 Unitary Similarity ........................................ 77
5.2 Normal Matrices—the Spectral Theorem ...................... 78
5.3 More about Normal Matrices ................................ 81
5.4 Conditions for Unitary Similarity ......................... 84
Exercises ...................................................... 86
Chapter 6. Hermitian Matrices .................................. 89
6.1 Conjugate Bilinear Forms .................................. 89
6.2 Properties of Hermitian Matrices and Inertia .............. 91
6.3 The Rayleigh-Ritz Ratio and the Courant-Fischer Theorem ... 94
6.4 Cauchy's Interlacing Theorem and Other Eigenvalue
Inequalities .............................................. 97
6.5 Positive Definite Matrices ................................ 99
6.6 Simultaneous Row and Column Operations ................... 102
6.7 Hadamard's Determinant Inequality ........................ 105
6.8 Polar Factorization and Singular Value Decomposition ..... 106
Exercises ..................................................... 109
Chapter 7. Vector and Matrix Norms ............................ 113
7.1 Vector Norms ............................................. 113
7.2 Matrix Norms ............................................. 117
Exercises ..................................................... 119
Chapter 8. Some Matrix Factorizations ......................... 121
8.1 Singular Value Decomposition ............................. 121
8.2 Householder Transformations .............................. 127
8.3 Using Householder Transformations to Get Triangular,
Hessenberg, and Tridiagonal Forms ........................ 129
8.4 Some Methods for Computing Eigenvalues ................... 134
8.5 LDU Factorization ........................................ 138
Exercises ..................................................... 141
Chapter 9. Field of Values .................................... 143
9.1 Basic Properties of the Field of Values .................. 143
9.2 The Field of Values for Two-by-Two Matrices .............. 145
9.3 Convexity of the Numerical Range ......................... 148
Exercises ..................................................... 150
Chapter 10. Simultaneous Triangularization .................... 151
10.1 Invariant Subspaces and Block Triangularization .......... 151
10.2 Simultaneous Triangularization, Property P, and
Commutativity ............................................ 152
10.3 Algebras, Ideals, and Nilpotent Ideals ................... 154
10.4 McCoy's Theorem .......................................... 157
10.5 Property L ............................................... 158
Exercises ..................................................... 161
Chapter 11. Circulant and Block Cycle Matrices ................ 163
11.1 The J Matrix ............................................. 163
11.2 Circulant Matrices ....................................... 163
11.3 Block Cycle Matrices ..................................... 165
Exercises ..................................................... 167
Chapter 12. Matrices of Zeros and Ones ........................ 169
12.1 Introduction: Adjacency Matrices and Incidence Matrices .. 169
12.2 Basic Facts about (0, 1)-Matrices ........................ 172
12.3 The Minimax Theorem of Kцnig and Egervдry ................ 173
12.4 SDRs, a Theorem of P. Hall, and Permanents ............... 174
12.5 Doubly Stochastic Matrices and Birkhoff's Theorem ........ 176
12.6 A Theorem of Ryser ....................................... 180
Exercises ..................................................... 182
Chapter 13. Block Designs ..................................... 185
13.1 t-Designs ................................................ 185
13.2 Incidence Matrices for 2-Designs ......................... 189
13.3 Finite Projective Planes ................................. 191
13.4 Quadratic Forms and the Witt Cancellation Theorem ........ 198
13.5 The Bruck-Ryser-Chowla Theorem ........................... 202
Exercises ..................................................... 205
Chapter 14. Hadamard Matrices ................................. 207
14.1 Introduction ............................................. 207
14.2 The Quadratic Residue Matrix and Paley's Theorem ......... 208
14.3 Results of Williamson .................................... 212
14.4 Hadamard Matrices and Block Designs ...................... 216
14.5 A Determinant Inequality, Revisited ...................... 219
Exercises ..................................................... 219
Chapter 15. Graphs ............................................ 221
15.1 Definitions .............................................. 221
15.2 Graphs and Matrices ...................................... 223
15.3 Walks and Cycles ......................................... 224
15.4 Graphs and Eigenvalues ................................... 226
15.5 Strongly Regular Graphs .................................. 227
Exercises ..................................................... 232
Chapter 16. Directed Graphs ................................... 235
16.1 Definitions .............................................. 235
16.2 Irreducibility and Strong Connectivity ................... 238
16.3 Index of Imprimitivity ................................... 242
16.4 Primitive Graphs ......................................... 245
Exercises ..................................................... 247
Chapter 17. Nonnegative Matrices ............................. 249
17.1 Introduction ............................................. 249
17.2 Preliminaries ............................................ 250
17.3 Proof of Perron's Theorem ................................ 254
17.4 Nonnegative Matrices ..................................... 258
17.5 Irreducible Matrices ..................................... 259
17.6 Primitive and Imprimitive Matrices ....................... 260
Exercises ..................................................... 262
Chapter 18. Error-Correcting Codes ............................ 265
18.1 Introduction ............................................. 265
18.2 The Hamming Code ......................................... 266
18.3 Linear Codes: Parity Check and Generator Matrices ........ 267
18.4 The Hamming Distance ..................................... 269
18.5 Perfect Codes and the Generalized Hamming Code ........... 271
18.6 Decoding ................................................. 273
18.7 Codes and Designs ........................................ 274
18.8 Hadamard Codes ........................................... 276
Exercises ..................................................... 277
Chapter 19. Linear Dynamical Systems .......................... 279
19.1 Introduction ............................................. 279
19.2 A Population Cohort Model ................................ 281
19.3 First-Order, Constant Coefficient, Linear Differential
and Difference Equations ................................. 283
19.4 Constant Coefficient, Homogeneous Systems ................ 285
19.5 Constant Coefficient, Nonhomogeneous Systems;
Equilibrium Points ....................................... 288
19.6 Nonnegative Systems ...................................... 292
19.7 Markov Chains ............................................ 295
Exercises ..................................................... 300
Bibliography .................................................. 303
Index ......................................................... 311
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