| Description |
xvii, 329 pages : illustrations ; 23 cm |
| Contents |
Contents note continued: 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 -- |
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Contents note continued: 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(An) wrt A is nA-1 -- |
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Contents note continued: 14.7.Differentiation of Determinants -- 14.7.1.Derivative of log(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 -- |
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Contents note continued: 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 -- |
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Contents note continued: 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 |
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Contents note continued: 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 -- |
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Contents note continued: 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 -- |
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Contents note continued: 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 -- |
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Contents note continued: 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 -- |
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Machine generated contents note: 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 -- |
| Bibliography |
Includes bibliographical references (pages 321-323) and index |
| Subject |
Matrices.
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R (Computer program language)
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| LC no. |
2011290290 |
| ISBN |
9789814313681 |
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9789814313698 |
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9814313688 |
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9814313696 |
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