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1 Contents Preface to Second Edition vii Preface to First Edition ix Part I Linear Algebra 1 Basic Vector/Matrix Structure and Notation Vectors Arrays Matrices Representation of Data What You Compute and What You Don t Vectors and Vector Spaces Operations on Vectors Linear Combinations and Linear Independence Vector Spaces and Spaces of Vectors Basis Sets for Vector Spaces Inner Products Norms Normalized Vectors Metrics and Distances Orthogonal Vectors and Orthogonal Vector Spaces The One Vector Cartesian Coordinates and Geometrical Properties of Vectors Cartesian Geometry Projections Angles between Vectors Orthogonalization Transformations; Gram-Schmidt Orthonormal Basis Sets
2 xviii Contents Approximation of Vectors Flats, Affine Spaces, and Hyperplanes Cones Cross Products in IR Centered Vectors and Variances and Covariances of Vectors The Mean and Centered Vectors The Standard Deviation, the Variance, andscaled Vectors Covariances and Correlations between Vectors Exercises Basic Properties of Matrices Basic Definitions and Notation Matrix Shaping Operators Partitioned Matrices Matrix Addition Scalar-Valued Operators on Square Matrices:The Trace Scalar-Valued Operators on Square Matrices:The Determinant Multiplication of Matrices and Multiplication ofvectors and Matrices Matrix Multiplication (Cayley) Multiplication of Matrices with Special Patterns Elementary Operations on Matrices The Trace of a Cayley Product that Is Square The Determinant of a Cayley Product of Square Matrices Multiplication of Matrices and Vectors Outer Products Bilinear and Quadratic Forms; Definiteness Anisometric Spaces Other Kinds of Matrix Multiplication Matrix Rank and the Inverse of a Matrix The Rank of Partitioned Matrices, Products of Matrices, and Sums of Matrices Full Rank Partitioning Full Rank Matrices and Matrix Inverses Full Rank Factorization Equivalent Matrices Multiplication by Full Rank Matrices Gramian Matrices: Products of the Form A T A A Lower Bound on the Rank of a Matrix Product Determinants of Inverses Inverses of Products and Sums of Nonsingular Matrices Inverses of Matrices with Special Forms Determining the Rank of a Matrix More on Partitioned Square Matrices: The Schur Complement Inverses of Partitioned Matrices Determinants of Partitioned Matrices Linear Systems of Equations
3 Contents xix Solutions of Linear Systems Null Space: The Orthogonal Complement Generalized Inverses Special Generalized Inverses; The Moore-Penrose Inverse Generalized Inverses of Products and Sums of Matrices Generalized Inverses of Partitioned Matrices Orthogonality Eigenanalysis; Canonical Factorizations Basic Properties of Eigenvalues and Eigenvectors The Characteristic Polynomial The Spectrum Similarity Transformations Schur Factorization Similar Canonical Factorization; Diagonalizable Matrices Properties of Diagonalizable Matrices Eigenanalysis of Symmetric Matrices Positive Definite and Nonnegative Definite Matrices Generalized Eigenvalues and Eigenvectors Singular Values and the Singular Value Decomposition (SVD) Matrix Norms Matrix Norms Induced from Vector Norms The Frobenius Norm The Usual Norm Other Matrix Norms Matrix Norm Inequalities The Spectral Radius Convergence of a Matrix Power Series Approximation of Matrices Exercises Vector/Matrix Derivatives and Integrals Basics of Differentiation Types of Differentiation Differentiation with Respect to a Scalar Differentiation with Respect to a Vector Differentiation with Respect to a Matrix Optimization of Scalar-Valued Functions Stationary Points of Functions Newton s Method Least Squares Maximum Likelihood Optimization of Functions with Constraints Optimization without Differentiation Integration and Expectation: Applications to Probability Distributions Multidimensional Integrals and Integrals InvolvingVectors and Matrices Integration Combined with Other Operations
4 xx Contents Random Variables and Probability Distributions Exercises Matrix Transformations and Factorizations Linear Geometric Transformations Transformations by Orthogonal Matrices Rotations Reflections Translations; Homogeneous Coordinates Householder Transformations (Reflections) Givens Transformations (Rotations) Factorization of Matrices LU and LDU Factorizations QR Factorization Householder Reflections to Form the QR Factorization Givens Rotations to Form the QR Factorization Gram-Schmidt Transformations to Form theqr Factorization Factorizations of Nonnegative Definite Matrices Square Roots Cholesky Factorization Factorizations of a Gramian Matrix Approximate Matrix Factorization Nonnegative Matrix Factorization Incomplete Factorizations Exercises Solution of Linear Systems Condition of Matrices Condition Number Improving the Condition Number Numerical Accuracy Direct Methods for Consistent Systems Gaussian Elimination and Matrix Factorizations Choice of Direct Method Iterative Methods for Consistent Systems The Gauss-Seidel Method withsuccessive Overrelaxation Conjugate Gradient Methods for SymmetricPositive Definite Systems Multigrid Methods Iterative Refinement Updating a Solution to a Consistent System Overdetermined Systems; Least Squares Least Squares Solution of an Overdetermined System Least Squares with a Full Rank Coefficient Matrix Least Squares with a Coefficient MatrixNot of Full Rank Weighted Least Squares
5 Contents xxi Updating a Least Squares Solution of anoverdetermined System Other Solutions of Overdetermined Systems Solutions that Minimize Other Norms of the Residuals Regularized Solutions Minimizing Orthogonal Distances Exercises Evaluation of Eigenvalues and Eigenvectors General Computational Methods Numerical Condition of an Eigenvalue Problem Eigenvalues from Eigenvectors and Vice Versa Deflation Preconditioning Shifting Power Method Jacobi Method QR Method Krylov Methods Generalized Eigenvalues Singular Value Decomposition Exercises Part II Applications in Data Analysis 8 Special Matrices and Operations Useful in Modeling anddata Analysis Data Matrices and Association Matrices Flat Files Graphs and Other Data Structures Term-by-Document Matrices Probability Distribution Models Derived Association Matrices Symmetric Matrices and Other Unitarily Diagonalizable Matrices Some Important Properties of Symmetric Matrices Approximation of Symmetric Matrices and an Important Inequality Normal Matrices Nonnegative Definite Matrices; Cholesky Factorization Positive Definite Matrices Idempotent and Projection Matrices Idempotent Matrices Projection Matrices: Symmetric Idempotent Matrices Special Matrices Occurring in Data Analysis Gramian Matrices Projection and Smoothing Matrices Centered Matrices and Variance-Covariance Matrices.. 363
6 xxii Contents The Generalized Variance Similarity Matrices Dissimilarity Matrices Nonnegative and Positive Matrices Properties of Square Positive Matrices Irreducible Square Nonnegative Matrices Stochastic Matrices Leslie Matrices Other Matrices with Special Structures Helmert Matrices Vandermonde Matrices Hadamard Matrices and Orthogonal Arrays Toeplitz Matrices Circulant Matrices Fourier Matrices and the Discrete Fourier Transform Hankel Matrices Cauchy Matrices Matrices Useful in Graph Theory Z-Matrices and M-Matrices Exercises Selected Applications in Statistics Multivariate Probability Distributions Basic Definitions and Properties The Multivariate Normal Distribution Derived Distributions and Cochran s Theorem Linear Models Fitting the Model Linear Models and Least Squares Statistical Inference The Normal Equations and the Sweep Operator Linear Least Squares Subject to LinearEquality Constraints Weighted Least Squares Updating Linear Regression Statistics Linear Smoothing Multivariate Linear Models Principal Components Principal Components of a Random Vector Principal Components of Data Condition of Models and Data Ill-Conditioning in Statistical Applications Variable Selection Principal Components Regression Shrinkage Estimation Statistical Inference about the Rank of a Matrix
7 Contents xxiii Incomplete Data Optimal Design Multivariate Random Number Generation Stochastic Processes Markov Chains Markovian Population Models Autoregressive Processes Exercises Part III Numerical Methods and Software 10 Numerical Methods Digital Representation of Numeric Data The Fixed-Point Number System The Floating-Point Model for Real Numbers Language Constructs for Representing Numeric Data Other Variations in the Representation of Data;Portability of Data Computer Operations on Numeric Data Fixed-Point Operations Floating-Point Operations Language Constructs for Operations onnumeric Data Software Methods for Extending the Precision Exact Computations Numerical Algorithms and Analysis Error in Numerical Computations Efficiency Iterations and Convergence Other Computational Techniques Exercises Numerical Linear Algebra Computer Storage of Vectors and Matrices General Computational Considerations forvectors and Matrices Relative Magnitudes of Operands Iterative Methods Assessing Computational Errors Multiplication of Vectors and Matrices Other Matrix Computations Exercises Software for Numerical Linear Algebra General Considerations Software Design Software Development, Maintenance, and Testing
8 xxiv Contents Reproducible Research Software Libraries BLAS Level 2 and Level 3 BLAS, LAPACK, and Related Libraries Libraries for High Performance Computing The IMSL Libraries General Purpose Languages Programming Considerations Modern Fortran C and C Python Interactive Systems for Array Manipulation R MATLAB and Octave Exercises Appendices and Back Matter A Notation and Definitions A.1 General Notation A.2 Computer Number Systems A.3 General Mathematical Functions and Operators A.4 Linear Spaces and Matrices A.5 Models and Data B Solutions and Hints for Selected Exercises Bibliography Index
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