LINEAR ALGEBRA AND MATRICES: TOPICS FOR A SECOND COURSE. VOLUME: 24

Linear algebra and matrix theory are fundamental tools for almost every area of mathematics, both pure and applied. This book combines coverage of core topics with an introduction to some areas in which linear algebra plays a key role, for example, block designs, directed graphs, error correcting codes, and linear dynamical systems. Notable features include a discussion of the Weyr characteristic and Weyr canonical forms, and their relationship to the better-known Jordan canonical form; the use of block cyclic matrices and directed graphs to prove Frobenius's theorem on the structure of the eigenvalues of a nonnegative, irreducible matrix; and the inclusion of such combinatorial topics as BIBDs, Hadamard matrices, and strongly regular graphs. Also included are McCoy's theorem about matrices with property P, the Bruck-Ryser-Chowla theorem on the existence of block designs, and an introduction to Markov chains. This book is intended for those who are familiar with the linear algebra covered in a typical first course and are interested in learning more advanced results.

Readership
Undergraduate and graduate students and research mathematicians interested in linear algebra, linear systems, graph theory, block designs, matrices, and error correcting codes.

Author
Helene Shapiro, Swarthmore College, PA

Contents
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 Cn 21
2.4. Orthogonal Complements and Projection onto a Subspace 23
2.5. Hilbert Spaces and Fourier Series 27
2.6. Unitary Tranformations 31
2.7. The Gram–Schmidt Process and QR Factorization 33
2.8. Linear Functionals 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 Ger?sgorin Circle Theorem 45
3.6. More about the Characteristic Polynomial 46
3.7. Eigenvalues of AB and BA 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 K¨onig and Egerv´ary 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