Hands-On Matrix Algebra Using R

The following sections are included:

• Preface

• Contents

The following sections are included:

• 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

The following sections are included:

• 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

The following sections are included:

• Vectors
  • Inner or Dot Product and Euclidean Length or Norm
  • Angle Between Two Vectors, Orthogonal Vectors

• Vector Spaces and Linear Operations
  • 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

The following sections are included:

• 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

The following sections are included:

• Payoff Matrix and Tools for Practical Decisions

• Maximax Solution

• Maximin Solution

• Minimax Regret Solution

• Digression: Mathematical Expectation from Vector Multiplication

• Maximum Expected Value Principle

• General R Function ‘payoff.all’ for Decisions

• Payoff Matrix in Job Search

The following sections are included:

• Cofactor of a Matrix

• Properties of Determinants

• Cramer's Rule and Ratios of Determinants

• Zero Determinant and Singularity
  • Nonsingularity

The following sections are included:

• 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

The following sections are included:

• 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
  • Simultaneous Equation Models
  • Haavelmo Model in Matrices
  • Population Growth Model from Demography

The following sections are included:

• 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

The following sections are included:

• 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

The following sections are included:

• 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

• Cholesky Decomposition

• Inequalities for Positive Definite Matrices

• Hadamard Product
  • Frobenius Product of Matrices

• Stochastic Matrices

• Ratios of Quadratic Forms, Rayleigh Quotient

The following sections are included:

• 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

The following sections are included:

• 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

The following sections are included:

• 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
  • Derivatives of a 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(Aⁿ) wrt A is nA⁻¹

• Differentiation of Determinants
  • Derivative of log(detA) wrt A is (A⁻¹)′

• Further Derivative Formulas for Vec and A⁻¹
  • Derivative of Matrix Inverse wrt Its Elements

• Optimization in Portfolio Choice Problem

The following sections are included:

• 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

The following sections are included:

• 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
  • 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

The following sections are included:

• 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

The following sections are included:

• Bibliography

• Index