Preface Note to the Reader Chapter 1.Preliminaries 1.1.Vector Spaces 1.2.Bases and Coordinates 1.3.Linear Transformations 1.4.Matrices 1.5.The Matrix of a Linear Transformation 1.6.Change of Basis and Similarity 1.7.Transposes 1.8.Special Types of Matrices 1.9.Submatrices, Partitioned Matrices, and Block Multiplication 1.10.Invariant Subspaces 1.11.Determinants 1.12.Tensor Products Exercises Chapter 2.Inner Product Spaces and Orthogonality 2.1.The Inner Product 2.2.Length, Orthogonality, and Projection onto a Line 2.3.Inner Products in C” 2.4.Orthogonal Complements and Projection onto a Subspace 2.5.Hilbert Spaces and Fourier Series 2.6.Unitary Tranformations 2.7.The Gram-Schmidt Process and QR Factorization 2.8.Linear Functionals and the Dual Space Exercises Chapter 3.Eigenvalues, Eigenvectors, Diagonalization, and Triangularization 3.1.Eigenvalues 3.2.Algebraic and Geometric Multiplicity 3.3.Diagonalizability 3.4.A Triangularization Theorem 3.5.The Gersgorin Circle Theorem 3.6.More about the Characteristic Polynomial 3.7.Eigenvalues of AB and BA Exercises Chapter 4.The Jordan and Weyr Canonical Forms 4.1.A Theorem of Sylvester and Reduction to Block Diagonal Form 4.2.Nilpotent Matrices 4.3.The Jordan Form of a General Matrix 4.4.The Cayley-Hamilton Theorem and the Minimal Polynomial 4.5.Weyr Normal Form Exercises Chapter 5.Unitary Similarity and Normal Matrices 5.1.Unitary Similarity 5.2.Normal Matrices—the Spectral Theorem 5.3.More about Normal Matrices 5.4.Conditions for Unitary Similarity Exercises Chapter 6.Hermitian Matrices 6.1.Conjugate Bilinear Forms 6.2.Properties of Hermitian Matrices and Inertia 6.3.The Rayleigh-Ritz Ratio and the Courant-Fischer Theorem 6.4.Cauchy's Interlacing Theorem and Other Eigenvalue Inequalities 6.5.Positive Definite Matrices 6.6.Simultaneous Row and Column Operations 6.7.Hadamard's Determinant Inequality 6.8.Polar Factorization and Singular Value Decomposition Exercises Chapter 7.Vector and Matrix Norms 7.1.Vector Norms 7.2.Matrix Norms Exercises Chapter 8.Some Matrix Factorizations 8.1.Singular Value Decomposition 8.2.Householder Transformations 8.3.Using Householder Transformations to Get Triangular, Hessenberg, and Tridiagonal Forms 8.4.Some Methods for Computing Eigenvalues 8.5.LDU Factorization Exercises Chapter 9.Field of Values 9.1.Basic Properties of the Field of Values 9.2.The Field of Values for Two-by-Two Matrices 9.3.Convexity of the Numerical Range Exercises Chapter 10.Simultaneous Triangularization 10.1.Invariant Subspaces and Block Triangularization 10.2.Simultaneous Triangularization, Property P, and Commutativity 10.3.Algebras, Ideals, and Nilpotent Ideals 10.4.McCoy's Theorem 10.5.Property L Exercises Chapter 11.Circulant and Block Cycle Matrices 11.1.The J Matrix 11.2.Circulant Matrices 11.3.Block Cycle Matrices Exercises Chapter 12.Matrices of Zeros and Ones 12.1.Introduction: Adjacency Matrices and Incidence Matrices 12.2.Basic Facts about (0, 1)-Matrices 12.3.The Minimax Theorem of K?nig and Egerváry 12.4.SDRs, a Theorem of P.Hall, and Permanents 12.5.Doubly Stochastic Matrices and Birkhoff's Theorem 12.6.A Theorem of Ryser Exercises Chapter 13.Block Designs 13.1.t-Designs 13.2.Incidence Matrices for 2-Designs 13.3.Finite Projective Planes 13.4.Quadratic Forms and the Witt Cancellation Theorem 13.5.The Bruck-Ryser-Chowla Theorem Exercises Chapter 14.Hadamard Matrices 14.1.Introduction 14.2.The Quadratic Residue Matrix and Paley's Theorem 14.3.Results of Williamson 14.4.Hadamard Matrices and Block Designs 14.5.A Determinant Inequality, Revisited Exercises Chapter 15.Graphs 15.1.Definitions 15.2.Graphs and Matrices 15.3.Walks and Cycles 15.4.Graphs and Eigenvalues
