Hands-on Matrix Algebra Using R

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1 Preface vii 1. R Preliminaries Matrix Defined, Deeper Understanding Using Software Introduction, Why R? Obtaining R Reference Manuals in R Basic R Language Tips Packages within R R Object Types and Their Attributes Dataframe Matrix and Its Summary Elementary Geometry and Algebra Using R Mathematical Functions Introductory Geometry and R Graphics Graphs for Simple Mathematical Functions and Equations Solving Linear Equation by Finding Roots Polyroot Function in R Bivariate Second Degree Equations and Their Plots Vector Spaces Vectors Inner or Dot Product and Euclidean Length or Norm Angle Between Two Vectors, Orthogonal Vectors Vector Spaces and Linear Operations xi

2 xii Hands-on Matrix Algebra Using R Linear Independence, Spanning and Basis Vector Space Defined Sum of Vectors in Vector Spaces Laws of Vector Algebra Column Space, Range Space and Null Space Transformations of Euclidean Plane Using Matrices Shrinkage and Expansion Maps Rotation Map Reflexion Maps Shifting the Origin or Translation Map Matrix to Compute Deviations from the Mean Projection in Euclidean Space Matrix Basics and R Software Matrix Notation Square Matrix Matrices Involving Complex Numbers Sum or Difference of Matrices Matrix Multiplication Transpose of a Matrix and Symmetric Matrices Reflexive Transpose Transpose of a Sum or Difference of Two Matrices Transpose of a Product of Two or More Matrices Symmetric Matrix Skew-symmetric Matrix Inner and Outer Products of Matrices Multiplication of a Matrix by a Scalar Multiplication of a Matrix by a Vector Further Rules for Sum and Product of Matrices Elementary Matrix Transformations Row Echelon Form LU Decomposition Decision Applications: Payoff Matrix Payoff Matrix and Tools for Practical Decisions Maximax Solution Maximin Solution Minimax Regret Solution

3 xiii 5.5 Digression: Mathematical Expectation from Vector Multiplication Maximum Expected Value Principle General R Function payoff.all for Decisions Payoff Matrix in Job Search Determinant and Singularity of a Square Matrix Cofactor of a Matrix Properties of Determinants Cramer s Rule and Ratios of Determinants Zero Determinant and Singularity Nonsingularity The Norm, Rank and Trace of a Matrix Norm of a Vector Cauchy-Schwartz Inequality Rank of a Matrix Properties of the Rank of a Matrix Trace of a Matrix Norm of a Matrix Matrix Inverse and Solution of Linear Equations Adjoint of a Matrix Matrix Inverse and Properties Matrix Inverse by Recursion Matrix Inversion When Two Terms Are Involved Solution of a Set of Linear Equations Ax = b Matrices in Solution of Difference Equations Matrix Inverse in Input-output Analysis Non-negativity in Matrix Algebra and Economics Diagonal Dominance Partitioned Matrices Sum and Product of Partitioned Matrices Block Triangular Matrix and Partitioned Matrix Determinant and Inverse Applications in Statistics and Econometrics Estimation of Heteroscedastic Variances MINQUE Estimator of Heteroscedastic Variances 151

4 xiv Hands-on Matrix Algebra Using R Simultaneous Equation Models Haavelmo Model in Matrices Population Growth Model from Demography Eigenvalues and Eigenvectors Characteristic Equation Eigenvectors n Eigenvalues n Eigenvectors Eigenvalues and Eigenvectors of Correlation Matrix Eigenvalue Properties Definite Matrices Eigenvalue-eigenvector Decomposition Orthogonal Matrix Idempotent Matrices Nilpotent and Tripotent matrices Similar Matrices, Quadratic and Jordan Canonical Forms Quadratic Forms Implying Maxima and Minima Positive, Negative and Other Definite Quadratic Forms Constrained Optimization and Bordered Matrices Bilinear Form Similar Matrices Diagonalizable Matrix Identity Matrix and Canonical Basis Generalized Eigenvectors and Chains Jordan Canonical Form Hermitian, Normal and Positive Definite Matrices Inner Product Admitting Complex Numbers Normal and Hermitian Matrices Real Symmetric and Positive Definite Matrices Square Root of a Matrix Positive Definite Hermitian Matrices Statistical Analysis of Variance and Quadratic Forms Second Degree Equation and Conic Sections

5 xv 11.4 Cholesky Decomposition Inequalities for Positive Definite Matrices Hadamard Product Frobenius Product of Matrices Stochastic Matrices Ratios of Quadratic Forms, Rayleigh Quotient Kronecker Products and Singular Value Decomposition Kronecker Product of Matrices Eigenvalues of Kronecker Products Eigenvectors of Kronecker Products Direct Sum of Matrices Singular Value Decomposition (SVD) SVD for Complex Number Matrices Condition Number of a Matrix Rule of Thumb for a Large Condition Number Pascal Matrix is Ill-conditioned Hilbert Matrix is Ill-conditioned Simultaneous Reduction and Vec Stacking Simultaneous Reduction of Two Matrices to a Diagonal Form Commuting Matrices Converting Matrices Into (Long) Vectors Vec of ABC Vec of (A + B) Trace of AB In Terms of Vec Trace of ABC In Terms of Vec Vech for Symmetric Matrices Vector and Matrix Differentiation Basics of Vector and Matrix Differentiation Chain Rule in Matrix Differentiation Chain Rule for Second Order Partials wrt θ Hessian Matrices in R Bordered Hessian for Utility Maximization Derivatives of Bilinear and Quadratic Forms Second Derivative of a Quadratic Form

6 xvi Hands-on Matrix Algebra Using R Derivatives of a Quadratic Form wrt θ Derivatives of a Symmetric Quadratic Form wrt θ Derivative of a Bilinear form wrt the Middle Matrix Derivative of a Quadratic Form wrt the Middle Matrix Differentiation of the Trace of a Matrix Derivatives of tr(ab), tr(abc) Derivative tr(a n ) wrt A is na Differentiation of Determinants Derivative of log(det A) wrt A is (A 1 ) Further Derivative Formulas for Vec and A Derivative of Matrix Inverse wrt Its Elements Optimization in Portfolio Choice Problem Matrix Results for Statistics Multivariate Normal Variables Bivariate Normal, Conditional Density and Regression Score Vector and Fisher Information Matrix Moments of Quadratic Forms in Normals Independence of Quadratic Forms Regression Applications of Quadratic Forms Vector Autoregression or VAR Models Canonical Correlations Taylor Series in Matrix Notation Generalized Inverse and Patterned Matrices Defining Generalized Inverse Properties of Moore-Penrose g-inverse Computation of g-inverse System of Linear Equations and Conditional Inverse Approximate Solutions to Inconsistent Systems Restricted Least Squares Vandermonde and Fourier Patterned Matrices Fourier Matrix Permutation Matrix Reducible matrix

7 xvii Nonnegative Indecomposable Matrices Perron-Frobenius Theorem Diagonal Band and Toeplitz Matrices Toeplitz Matrices Circulant Matrices Hankel Matrices Hadamard Matrices Mathematical Programming and Matrix Algebra Control Theory Applications of Matrix Algebra Brief Introduction to State Space Models Linear Quadratic Gaussian Problems Smoothing Applications of Matrix Algebra Numerical Accuracy and QR Decomposition Rounding Numbers Binary Arithmetic and Computer Bits Floating Point Arithmetic Fibonacci Numbers Using Matrices and Digital Computers Numerically More Reliable Algorithms Gram-Schmidt Orthogonalization The QR Modification of Gram-Schmidt QR Decomposition QR Algorithm Schur Decomposition Bibliography 321 Index 325

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