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Hands-on matrix algebra using R: active and motivated learning with applications
Author
Publisher
World Scientific
Publication Date
c2011
Language
English
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Table of Contents
From the Book
Preface
1. R Preliminaries
1.1. Matrix Defined, Deeper Understanding Using Software
1.2. Introduction, Why R?
1.3. Obtaining R
1.4. Reference Manuals in R
1.5. Basic R Language Tips
1.6. Packages within R
1.7. R Object Types and Their Attributes
1.7.1. Dataframe Matrix and Its Summary
2. Elementary Geometry and Algebra Using R
2.1. Mathematical Functions
2.2. Introductory Geometry and R Graphics
2.2.1. Graphs for Simple Mathematical Functions and Equations
2.3. Solving Linear Equation by Finding Roots
2.4. Polyroot Function in R
2.5. Bivariate Second Degree Equations and Their Plots
3. Vector Spaces
3.1. Vectors
3.1.1. Inner or Dot Product and Euclidean Length or Norm
3.1.2. Angle Between Two Vectors, Orthogonal Vectors
3.2. Vector Spaces and Linear Operations
3.2.1. Linear Independence, Spanning and Basis
3.2.2. Vector Space Defined
3.3. Sum of Vectors in Vector Spaces
3.3.1. Laws of Vector Algebra
3.3.2. Column Space, Range Space and Null Space
3.4. Transformations of Euclidean Plane Using Matrices
3.4.1. Shrinkage and Expansion Maps
3.4.2. Rotation Map
3.4.3. Reflexion Maps
3.4.4. Shifting the Origin or Translation Map
3.4.5. Matrix to Compute Deviations from the Mean
3.4.6. Projection in Euclidean Space
4. Matrix Basics and R Software
4.1. Matrix Notation
4.1.1. Square Matrix
4.2. Matrices Involving Complex Numbers
4.3. Sum or Difference of Matrices
4.4. Matrix Multiplication
4.5. Transpose of a Matrix and Symmetric Matrices
4.5.1. Reflexive Transpose
4.5.2. Transpose of a Sum or Difference of Two Matrices
4.5.3. Transpose of a Product of Two or More Matrices
4.5.4. Symmetric Matrix
4.5.5. Skew-symmetric Matrix
4.5.6. Inner and Outer Products of Matrices
4.6. Multiplication of a Matrix by a Scalar
4.7. Multiplication of a Matrix by a Vector
4.8. Further Rules for Sum and Product of Matrices
4.9. Elementary Matrix Transformations
4.9.1. Row Echelon Form
4.10. LU Decomposition
5. Decision Applications: Payoff Matrix
5.1. Payoff Matrix and Tools for Practical Decisions
5.2. Maximax Solution
5.3. Maximin Solution
5.4. Minimax Regret Solution
5.5. Digression: Mathematical Expectation from Vector Multiplication
5.6. Maximum Expected Value Principle
5.7. General R Function 'payoff.all for Decisions
5.8. Payoff Matrix in Job Search
6. Determinant and Singularity of a Square Matrix
6.1. Cofactor of a Matrix
6.2. Properties of Determinants
6.3. Cramer's Rule and Ratios of Determinants
6.4. Zero Determinant and Singularity
6.4.1. Nonsingularity
7. The Norm, Rank and Trace of a Matrix
7.1. Norm of a Vector
7.1.1. Cauchy-Schwartz Inequality
7.2. Rank of a Matrix
7.3. Properties of the Rank of a Matrix
7.4. Trace of a Matrix
7.5. Norm of a Matrix
8. Matrix Inverse and Solution of Linear Equations
8.1. Adjoint of a Matrix
8.2. Matrix Inverse and Properties
8.3. Matrix Inverse by Recursion
8.4. Matrix Inversion When Two Terms Are Involved
8.5. Solution of a Set of Linear Equations Ax = b
8.6. Matrices in Solution of Difference Equations
8.7. Matrix Inverse in Input-output Analysis
8.7.1. Non-negativity in Matrix Algebra and Economics
8.7.2. Diagonal Dominance
8.8. Partitioned Matrices
8.8.1. Sum and Product of Partitioned Matrices
8.8.2. Block Triangular Matrix and Partitioned Matrix Determinant and Inverse
8.9. Applications in Statistics and Econometrics
8.9.1. Estimation of Heteroscedastic Variances
8.9.2. Minque Estimator of Heteroscedastic Variances
8.9.3. Simultaneous Equation Models
8.9.4. Haavelmo Model in Matrices
8.9.5. Population Growth Model from Demography
9. Eigenvalues and Eigenvectors
9.1. Characteristic Equation
9.1.1. Eigenvectors
9.1.2. n Eigenvalues
9.1.3. n Eigenvectors
9.2. Eigenvalues and Eigenvectors of Correlation Matrix
9.3. Eigenvalue Properties
9.4. Definite Matrices
9.5. Eigenvalue-eigenvector Decomposition
9.5.1. Orthogonal Matrix
9.6. Idempotent Matrices
9.7. Nilpotent and Tripotent matrices
10. Similar Matrices, Quadratic and Jordan Canonical Forms
10.1. Quadratic Forms Implying Maxima and Minima
10.1.1. Positive, Negative and Other Definite Quadratic Forms
10.2. Constrained Optimization and Bordered Matrices
10.3. Bilinear Form
10.4. Similar Matrices
10.4.1. Diagonalizable Matrix
10.5. Identity Matrix and Canonical Basis
10.6. Generalized Eigenvectors and Chains
10.7. Jordan Canonical Form
11. Hermitian, Normal and Positive Definite Matrices
11.1. Inner Product Admitting Complex Numbers
11.2. Normal and Hermitian Matrices
11.3. Real Symmetric and Positive Definite Matrices
11.3.1. Square Root of a Matrix
11.3.2. Positive Definite Hermitian Matrices
11.3.3. Statistical Analysis of Variance and Quadratic Forms
11.3.4. Second Degree Equation and Conic Sections
11.4. Cholesky Decomposition
11.5. Inequalities for Positive Definite Matrices
11.6. Hadamard Product
11.6.1. Frobenius Product of Matrices
11.7. Stochastic Matrices
11.8. Ratios of Quadratic Forms, Rayleigh Quotient
12. Kronecker Products and Singular Value Decomposition
12.1. Kronecker Product of Matrices
12.1.1. Eigenvalues of Kronecker Products
12.1.2. Eigenvectors of Kronecker Products
12.1.3. Direct Sum of Matrices
12.2. Singular Value Decomposition (SVD)
12.2.1. SVD for Complex Number Matrices
12.3. Condition Number of a Matrix
12.3.1. Rule of Thumb for a Large Condition Number
12.3.2. Pascal Matrix is Ill-conditioned
12.4. Hilbert Matrix is Ill-conditioned
13. Simultaneous Reduction and Vec Stacking
13.1. Simultaneous Reduction of Two Matrices to a Diagonal Form
13.2. Commuting Matrices
13.3. Converting Matrices Into (Long) Vectors
13.3.1. Vec of ABC
13.3.2. Vec of (A+ B)
13.3.3. Trace of AB In Terms of Vec
13.3.4. Trace of ABC In Terms of Vec
13.4. Vech for Symmetric Matrices
14. Vector and Matrix Differentiation,
14.1. Basics of Vector and Matrix Differentiation
14.2. Chain Rule in Matrix Differentiation
14.2.1. Chain Rule for Second Order Partials wrt θ
14.2.2. Hessian Matrices in R
14.2.3. Bordered Hessian for Utility Maximization
14.3. Derivatives of Bilinear and Quadratic Forms
14.4. Second Derivative of a Quadratic Form
14.4.1. Derivatives of a Quadratic Form wrt θ
14.4.2. Derivatives of a Symmetric Quadratic Form wrt θ
14.4.3. Derivative of a Bilinear form wrt the Middle Matrix
14.4.4. Derivative of a Quadratic Form wrt the Middle Matrix
14.5. Differentiation of the Trace of a Matrix
14.6. Derivatives of tr(AB), tr(ABC)
14.6.1. Derivative tr(A n ) wrt A is nA -1
14.7. Differentiation of Determinants
14.7.1. Derivative of iog(det A) wrt A is (A -1 )
14.8. Further Derivative Formulas for Vec and A -1
14.8.1. Derivative of Matrix Inverse wrt Its Elements
14.9. Optimization in Portfolio Choice Problem
15. Matrix Results for Statistics
15.1. Multivariate Normal Variables
15.1.1. Bivariate Normal, Conditional Density and Regression
15.1.2. Score Vector and Fisher Information Matrix
15.2. Moments of Quadratic Forms in Normals
15.2.1. Independence of Quadratic Forms
15.3. Regression Applications of Quadratic Forms
15.4. Vector Autoregression or VAR Models
15.4.1. Canonical Correlations
15.5. Taylor Series in Matrix Notation
16. Generalized Inverse and Patterned Matrices
16.1. Defining Generalized Inverse
16.2. Properties of Moore-Penrose g-inverse
16.2.1. Computation of g-inverse
16.3. System of Linear Equations and Conditional Inverse
16.3.1. Approximate Solutions to Inconsistent Systems
16.3.2. Restricted Least Squares
16.4. Vandermonde and Fourier Patterned Matrices
16.4.1. Fourier Matrix
16.4.2. Permutation Matrix
16.4.3. Reducible matrix
16.4.4. Nonnegative Indecomposable Matrices
16.4.5. Perron-Frobenius Theorem
16.5. Diagonal Band and Toeplitz Matrices
16.5.1. Toeplitz Matrices
16.5.2. Circulant Matrices
16.5.3. Hankel Matrices
16.5.4. Hadamard Matrices
16.6. Mathematical Programming and Matrix Algebra
16.7. Control Theory Applications of Matrix Algebra
16.7.1. Brief Introduction to State Space Models
16.7.2. Linear Quadratic Gaussian Problems
16.8. Smoothing Applications of Matrix Algebra
17. Numerical Accuracy and QR Decomposition
17.1. Rounding Numbers
17.1.1. Binary Arithmetic and Computer Bits
17.1.2. Floating Point Arithmetic
17.1.3. Fibonacci Numbers Using Matrices and Digital Computers
17.2. Numerically More Reliable Algorithms
17.3. Gram-Schmidt Orthogonalization
17.4. The QR Modification of Gram-Schmidt
17.4.1. QR Decomposition
17.4.2. QR Algorithm
17.5. Schur Decomposition
Bibliography
Index
Author Notes
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ISBN
9789814313698
9789814313681
9789814313681
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