Contents vii 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

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