Book description
A brand new, fully updated edition of a popular classic on matrix differential calculus with applications in statistics and econometrics
This exhaustive, self-contained book on matrix theory and matrix differential calculus provides a treatment of matrix calculus based on differentials and shows how easy it is to use this theory once you have mastered the technique. Jan Magnus, who, along with the late Heinz Neudecker, pioneered the theory, develops it further in this new edition and provides many examples along the way to support it.
Matrix calculus has become an essential tool for quantitative methods in a large number of applications, ranging from social and behavioral sciences to econometrics. It is still relevant and used today in a wide range of subjects such as the biosciences and psychology. Matrix Differential Calculus with Applications in Statistics and Econometrics, Third Edition contains all of the essentials of multivariable calculus with an emphasis on the use of differentials. It starts by presenting a concise, yet thorough overview of matrix algebra, then goes on to develop the theory of differentials. The rest of the text combines the theory and application of matrix differential calculus, providing the practitioner and researcher with both a quick review and a detailed reference.
- Fulfills the need for an updated and unified treatment of matrix differential calculus
- Contains many new examples and exercises based on questions asked of the author over the years
- Covers new developments in field and features new applications
- Written by a leading expert and pioneer of the theory
- Part of the Wiley Series in Probability and Statistics
Matrix Differential Calculus With Applications in Statistics and Econometrics Third Edition is an ideal text for graduate students and academics studying the subject, as well as for postgraduates and specialists working in biosciences and psychology.
Table of contents
- Cover
- Preface
-
Part One: Matrices
-
Chapter 1: Basic properties of vectors and matrices
- 1 INTRODUCTION
- 2 SETS
- 3 MATRICES: ADDITION AND MULTIPLICATION
- 4 THE TRANSPOSE OF A MATRIX
- 5 SQUARE MATRICES
- 6 LINEAR FORMS AND QUADRATIC FORMS
- 7 THE RANK OF A MATRIX
- 8 THE INVERSE
- 9 THE DETERMINANT
- 10 THE TRACE
- 11 PARTITIONED MATRICES
- 12 COMPLEX MATRICES
- 13 EIGENVALUES AND EIGENVECTORS
- 14 SCHUR'S DECOMPOSITION THEOREM
- 15 THE JORDAN DECOMPOSITION
- 16 THE SINGULAR‐VALUE DECOMPOSITION
- 17 FURTHER RESULTS CONCERNING EIGENVALUES
- 18 POSITIVE (SEMI)DEFINITE MATRICES
- 19 THREE FURTHER RESULTS FOR POSITIVE DEFINITE MATRICES
- 20 A USEFUL RESULT
- 21 SYMMETRIC MATRIX FUNCTIONS
- Chapter 2: Kronecker products, vec operator, and Moore‐Penrose inverse
-
Chapter 3: Miscellaneous matrix results
- 1 INTRODUCTION
- 2 THE ADJOINT MATRIX
- 3 PROOF OF THEOREM 3.1
- 4 BORDERED DETERMINANTS
- 5 THE MATRIX EQUATION AX = 0
- 6 THE HADAMARD PRODUCT
- 7 THE COMMUTATION MATRIX Kmn
- 8 THE DUPLICATION MATRIX Dn
- 9 RELATIONSHIP BETWEEN Dn +1 AND Dn , I
- 10 RELATIONSHIP BETWEEN Dn +1 AND Dn , II
- 11 CONDITIONS FOR A QUADRATIC FORM TO BE POSITIVE (NEGATIVE) SUBJECT TO LINEAR CONSTRAINTS
- 12 NECESSARY AND SUFFICIENT CONDITIONS FOR r(A : B) = r(A) + r(B)
- 13 THE BORDERED GRAMIAN MATRIX
- 14 THE EQUATIONS X 1 A + X 2 B′ = G 1, X 1 B = G 2
- BIBLIOGRAPHICAL NOTES
-
Chapter 1: Basic properties of vectors and matrices
-
Part Two: Differentials: the theory
- Chapter 4: Mathematical preliminaries
-
Chapter 5: Differentials and differentiability
- 1 INTRODUCTION
- 2 CONTINUITY
- 3 DIFFERENTIABILITY AND LINEAR APPROXIMATION
- 4 THE DIFFERENTIAL OF A VECTOR FUNCTION
- 5 UNIQUENESS OF THE DIFFERENTIAL
- 6 CONTINUITY OF DIFFERENTIABLE FUNCTIONS
- 7 PARTIAL DERIVATIVES
- 8 THE FIRST IDENTIFICATION THEOREM
- 9 EXISTENCE OF THE DIFFERENTIAL, I
- 10 EXISTENCE OF THE DIFFERENTIAL, II
- 11 CONTINUOUS DIFFERENTIABILITY
- 12 THE CHAIN RULE
- 13 CAUCHY INVARIANCE
- 14 THE MEAN‐VALUE THEOREM FOR REAL‐VALUED FUNCTIONS
- 15 DIFFERENTIABLE MATRIX FUNCTIONS
- 16 SOME REMARKS ON NOTATION
- 17 COMPLEX DIFFERENTIATION
- BIBLIOGRAPHICAL NOTES
-
Chapter 6: The second differential
- 1 INTRODUCTION
- 2 SECOND‐ORDER PARTIAL DERIVATIVES
- 3 THE HESSIAN MATRIX
- 4 TWICE DIFFERENTIABILITY AND SECOND‐ORDER APPROXIMATION, I
- 5 DEFINITION OF TWICE DIFFERENTIABILITY
- 6 THE SECOND DIFFERENTIAL
- 7 SYMMETRY OF THE HESSIAN MATRIX
- 8 THE SECOND IDENTIFICATION THEOREM
- 9 TWICE DIFFERENTIABILITY AND SECOND‐ORDER APPROXIMATION, II
- 10 CHAIN RULE FOR HESSIAN MATRICES
- 11 THE ANALOG FOR SECOND DIFFERENTIALS
- 12 TAYLOR'S THEOREM FOR REAL‐VALUED FUNCTIONS
- 13 HIGHER‐ORDER DIFFERENTIALS
- 14 REAL ANALYTIC FUNCTIONS
- 15 TWICE DIFFERENTIABLE MATRIX FUNCTIONS
- BIBLIOGRAPHICAL NOTES
-
Chapter 7: Static optimization
- 1 INTRODUCTION
- 2 UNCONSTRAINED OPTIMIZATION
- 3 THE EXISTENCE OF ABSOLUTE EXTREMA
- 4 NECESSARY CONDITIONS FOR A LOCAL MINIMUM
- 5 SUFFICIENT CONDITIONS FOR A LOCAL MINIMUM: FIRST‐DERIVATIVE TEST
- 6 SUFFICIENT CONDITIONS FOR A LOCAL MINIMUM: SECOND‐DERIVATIVE TEST
- 7 CHARACTERIZATION OF DIFFERENTIABLE CONVEX FUNCTIONS
- 8 CHARACTERIZATION OF TWICE DIFFERENTIABLE CONVEX FUNCTIONS
- 9 SUFFICIENT CONDITIONS FOR AN ABSOLUTE MINIMUM
- 10 MONOTONIC TRANSFORMATIONS
- 11 OPTIMIZATION SUBJECT TO CONSTRAINTS
- 12 NECESSARY CONDITIONS FOR A LOCAL MINIMUM UNDER CONSTRAINTS
- 13 SUFFICIENT CONDITIONS FOR A LOCAL MINIMUM UNDER CONSTRAINTS
- 14 SUFFICIENT CONDITIONS FOR AN ABSOLUTE MINIMUM UNDER CONSTRAINTS
- 15 A NOTE ON CONSTRAINTS IN MATRIX FORM
- 16 ECONOMIC INTERPRETATION OF LAGRANGE MULTIPLIERS
- APPENDIX: THE IMPLICIT FUNCTION THEOREM
-
Part Three: Differentials: the practice
-
Chapter 8: Some important differentials
- 1 INTRODUCTION
- 2 FUNDAMENTAL RULES OF DIFFERENTIAL CALCULUS
- 3 THE DIFFERENTIAL OF A DETERMINANT
- 4 THE DIFFERENTIAL OF AN INVERSE
- 5 DIFFERENTIAL OF THE MOORE‐PENROSE INVERSE
- 6 THE DIFFERENTIAL OF THE ADJOINT MATRIX
- 7 ON DIFFERENTIATING EIGENVALUES AND EIGENVECTORS
- 8 THE CONTINUITY OF EIGENPROJECTIONS
- 9 THE DIFFERENTIAL OF EIGENVALUES AND EIGENVECTORS: SYMMETRIC CASE
- 10 TWO ALTERNATIVE EXPRESSIONS FOR dλ
- 11 SECOND DIFFERENTIAL OF THE EIGENVALUE FUNCTION
-
Chapter 9: First‐order differentials and Jacobian matrices
- 1 INTRODUCTION
- 2 CLASSIFICATION
- 3 DERISATIVES
- 4 DERIVATIVES
- 5 IDENTIFICATION OF JACOBIAN MATRICES
- 6 THE FIRST IDENTIFICATION TABLE
- 7 PARTITIONING OF THE DERIVATIVE
- 8 SCALAR FUNCTIONS OF A SCALAR
- 9 SCALAR FUNCTIONS OF A VECTOR
- 10 SCALAR FUNCTIONS OF A MATRIX, I: TRACE
- 11 SCALAR FUNCTIONS OF A MATRIX, II: DETERMINANT
- 12 SCALAR FUNCTIONS OF A MATRIX, III: EIGENVALUE
- 13 TWO EXAMPLES OF VECTOR FUNCTIONS
- 14 MATRIX FUNCTIONS
- 15 KRONECKER PRODUCTS
- 16 SOME OTHER PROBLEMS
- 17 JACOBIANS OF TRANSFORMATIONS
- Chapter 10: Second‐order differentials and Hessian matrices
-
Chapter 8: Some important differentials
-
Part Four: Inequalities
-
Chapter 11: Inequalities
- 1 INTRODUCTION
- 2 THE CAUCHY‐SCHWARZ INEQUALITY
- 3 MATRIX ANALOGS OF THE CAUCHY‐SCHWARZ INEQUALITY
- 4 THE THEOREM OF THE ARITHMETIC AND GEOMETRIC MEANS
- 5 THE RAYLEIGH QUOTIENT
- 6 CONCAVITY OF λ 1 AND CONVEXITY OF λ n
- 7 VARIATIONAL DESCRIPTION OF EIGENVALUES
- 8 FISCHER'S MIN‐MAX THEOREM
- 9 MONOTONICITY OF THE EIGENVALUES
- 10 THE POINCARÉ SEPARATION THEOREM
- 11 TWO COROLLARIES OF POINCARÉ'S THEOREM
- 12 FURTHER CONSEQUENCES OF THE POINCARÉ THEOREM
- 13 MULTIPLICATIVE VERSION
- 14 THE MAXIMUM OF A BILINEAR FORM
- 15 HADAMARD'S INEQUALITY
- 16 AN INTERLUDE: KARAMATA'S INEQUALITY
- 17 KARAMATA'S INEQUALITY AND EIGENVALUES
- 18 AN INEQUALITY CONCERNING POSITIVE SEMIDEFINITE MATRICES
- 19 A REPRESENTATION THEOREM FOR
- 20 A REPRESENTATION THEOREM FOR (tr A)
- 21 HÖLDER'S INEQUALITY
- 22 CONCAVITY OF log|A|
- 23 MINKOWSKI'S INEQUALITY
- 24 QUASILINEAR REPRESENTATION OF |A|
- 25 MINKOWSKI'S DETERMINANT THEOREM
- 26 WEIGHTED MEANS OF ORDER p
- 27 SCHLÖMILCH'S INEQUALITY
- 28 CURVATURE PROPERTIES OF Mp (x, a)
- 29 LEAST SQUARES
- 30 GENERALIZED LEAST SQUARES
- 31 RESTRICTED LEAST SQUARES
- 32 RESTRICTED LEAST SQUARES: MATRIX VERSION
-
Chapter 11: Inequalities
-
Part Five: The linear model
-
Chapter 12: Statistical preliminaries
- 1 INTRODUCTION
- 2 THE CUMULATIVE DISTRIBUTION FUNCTION
- 3 THE JOINT DENSITY FUNCTION
- 4 EXPECTATIONS
- 5 VARIANCE AND COVARIANCE
- 6 INDEPENDENCE OF TWO RANDOM VARIABLES
- 7 INDEPENDENCE OF n RANDOM VARIABLES
- 8 SAMPLING
- 9 THE ONE‐DIMENSIONAL NORMAL DISTRIBUTION
- 10 THE MULTIVARIATE NORMAL DISTRIBUTION
- 11 ESTIMATION
-
Chapter 13: The linear regression model
- 1 INTRODUCTION
- 2 AFFINE MINIMUM‐TRACE UNBIASED ESTIMATION
- 3 THE GAUSS‐MARKOV THEOREM
- 4 THE METHOD OF LEAST SQUARES
- 5 AITKEN'S THEOREM
- 6 MULTICOLLINEARITY
- 7 ESTIMABLE FUNCTIONS
- 8 LINEAR CONSTRAINTS: THE CASEℳ(R) ⊂ ℳ(X)
- 9 LINEAR CONSTRAINTS: THE GENERAL CASE
- 10 LINEAR CONSTRAINTS: THE CASEℳ(R) ∩ ℳ(X) = {0}
- 11 A SINGULAR VARIANCE MATRIX: THE CASEℳ(X) ⊂ ℳ(V)
- 12 A SINGULAR VARIANCE MATRIX: THE CASE r(X′V X) = r(X)
- 13 A SINGULAR VARIANCE MATRIX: THE GENERAL CASE, I
- 14 EXPLICIT AND IMPLICIT LINEAR CONSTRAINTS
- 15 THE GENERAL LINEAR MODEL, I
- 16 A SINGULAR VARIANCE MATRIX: THE GENERAL CASE, II
- 17 THE GENERAL LINEAR MODEL, II
- 18 GENERALIZED LEAST SQUARES
- 19 RESTRICTED LEAST SQUARES
-
Chapter 14: Further topics in the linear model
- 1 INTRODUCTION
- 2 BEST QUADRATIC UNBIASED ESTIMATION OF σ
- 3 THE BEST QUADRATIC AND POSITIVE UNBIASED ESTIMATOR OF σ
- 4 THE BEST QUADRATIC UNBIASED ESTIMATOR OF σ
- 5 BEST QUADRATIC INVARIANT ESTIMATION OF σ
- 6 THE BEST QUADRATIC AND POSITIVE INVARIANT ESTIMATOR OF σ
- 7 THE BEST QUADRATIC INVARIANT ESTIMATOR OF σ
- 8 BEST QUADRATIC UNBIASED ESTIMATION: MULTIVARIATE NORMAL CASE
- 9 BOUNDS FOR THE BIAS OF THE LEAST‐SQUARES ESTIMATOR OF σ, I
- 10 BOUNDS FOR THE BIAS OF THE LEAST‐SQUARES ESTIMATOR OF σ, II
- 11 THE PREDICTION OF DISTURBANCES
- 12 BEST LINEAR UNBIASED PREDICTORS WITH SCALAR VARIANCE MATRIX
- 13 BEST LINEAR UNBIASED PREDICTORS WITH FIXED VARIANCE MATRIX, I
- 14 BEST LINEAR UNBIASED PREDICTORS WITH FIXED VARIANCE MATRIX, II
- 15 LOCAL SENSITIVITY OF THE POSTERIOR MEAN
- 16 LOCAL SENSITIVITY OF THE POSTERIOR PRECISION
-
Chapter 12: Statistical preliminaries
-
Part Six: Applications to maximum likelihood estimation
-
Chapter 15: Maximum likelihood estimation
- 1 INTRODUCTION
- 2 THE METHOD OF MAXIMUM LIKELIHOOD (ML)
- 3 ML ESTIMATION OF THE MULTIVARIATE NORMAL DISTRIBUTION
- 4 SYMMETRY: IMPLICIT VERSUS EXPLICIT TREATMENT
- 5 THE TREATMENT OF POSITIVE DEFINITENESS
- 6 THE INFORMATION MATRIX
- 7 ML ESTIMATION OF THE MULTIVARIATE NORMAL DISTRIBUTION: DISTINCT MEANS
- 8 THE MULTIVARIATE LINEAR REGRESSION MODEL
- 9 THE ERRORS‐IN‐VARIABLES MODEL
- 10 THE NONLINEAR REGRESSION MODEL WITH NORMAL ERRORS
- 11 SPECIAL CASE: FUNCTIONAL INDEPENDENCE OF MEAN AND VARIANCE PARAMETERS
- 12 GENERALIZATION OF THEOREM 15.6
-
Chapter 16: Simultaneous equations
- 1 INTRODUCTION
- 2 THE SIMULTANEOUS EQUATIONS MODEL
- 3 THE IDENTIFICATION PROBLEM
- 4 IDENTIFICATION WITH LINEAR CONSTRAINTS ON B AND Γ ONLY
- 5 IDENTIFICATION WITH LINEAR CONSTRAINTS ON B, Γ, AND Σ
- 6 NONLINEAR CONSTRAINTS
- 7 FIML: THE INFORMATION MATRIX (GENERAL CASE)
- 8 FIML: ASYMPTOTIC VARIANCE MATRIX (SPECIAL CASE)
- 9 LIML: FIRST‐ORDER CONDITIONS
- 10 LIML: INFORMATION MATRIX
- 11 LIML: ASYMPTOTIC VARIANCE MATRIX
- BIBLIOGRAPHICAL NOTES
-
Chapter 17: Topics in psychometrics
- 1 INTRODUCTION
- 2 POPULATION PRINCIPAL COMPONENTS
- 3 OPTIMALITY OF PRINCIPAL COMPONENTS
- 4 A RELATED RESULT
- 5 SAMPLE PRINCIPAL COMPONENTS
- 6 OPTIMALITY OF SAMPLE PRINCIPAL COMPONENTS
- 7 ONE‐MODE COMPONENT ANALYSIS
- 8 ONE‐MODE COMPONENT ANALYSIS AND SAMPLE PRINCIPAL COMPONENTS
- 9 TWO‐MODE COMPONENT ANALYSIS
- 10 MULTIMODE COMPONENT ANALYSIS
- 11 FACTOR ANALYSIS
- 12 A ZIGZAG ROUTINE
- 13 NEWTON‐RAPHSON ROUTINE
- 14 KAISER'S VARIMAX METHOD
- 15 CANONICAL CORRELATIONS AND VARIATES IN THE POPULATION
- 16 CORRESPONDENCE ANALYSIS
- 17 LINEAR DISCRIMINANT ANALYSIS
- BIBLIOGRAPHICAL NOTES
-
Chapter 15: Maximum likelihood estimation
-
Part Seven: Summary
-
Chapter 18: Matrix calculus: the essentials
- 1 INTRODUCTION
- 2 DIFFERENTIALS
- 3 VECTOR CALCULUS
- 4 OPTIMIZATION
- 5 LEAST SQUARES
- 6 MATRIX CALCULUS
- 7 INTERLUDE ON LINEAR AND QUADRATIC FORMS
- 8 THE SECOND DIFFERENTIAL
- 9 CHAIN RULE FOR SECOND DIFFERENTIALS
- 10 FOUR EXAMPLES
- 11 THE KRONECKER PRODUCT AND VEC OPERATOR
- 12 IDENTIFICATION
- 13 THE COMMUTATION MATRIX
- 14 FROM SECOND DIFFERENTIAL TO HESSIAN
- 15 SYMMETRY AND THE DUPLICATION MATRIX
- 16 MAXIMUM LIKELIHOOD
- FURTHER READING
-
Chapter 18: Matrix calculus: the essentials
- Bibliography
- Index of symbols
- Subject index
- End User License Agreement
Product information
- Title: Matrix Differential Calculus with Applications in Statistics and Econometrics, 3rd Edition
- Author(s):
- Release date: March 2019
- Publisher(s): Wiley
- ISBN: 9781119541202
You might also like
book
Linear Algebra and Matrix Analysis for Statistics
Assuming no prior knowledge of linear algebra, this self-contained text offers a gradual exposition to linear …
book
Mathematical Statistics and Stochastic Processes
Generally, books on mathematical statistics are restricted to the case of independent identically distributed random variables. …
book
Calculus II
Idiot's Guides: Calculus II, like its counterpart Idiot's Guides: Calculus I, is a curriculum-based companion book …
book
Precalculus with Calculus Previews, 6th Edition
Building off the success of Zill and Dewar's popular Essentials version, the new Sixth Edition of …