Matrix differential calculus with applications in statistics and econometrics / Jan Rudolph Magnus and Heinz Neudecker.

Format:

Book

Author/Creator:

Magnus, Jan R., author.

Neudecker, Heinz, author.

Series:

Wiley series in probability and statistics.

Wiley Series in Probability and Statistics

Language:

English

Subjects (All):

Matrices.

Differential calculus.

Statistics.

Econometrics.

Physical Description:

1 online resource (500 pages).

Edition:

Third edition.

Place of Publication:

Hoboken, NJ : Wiley, 2019.

System Details:

text file

Summary:

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.

Contents:

Cover

Title Page

Copyright

Contents

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

Miscellaneous exercises

Bibliographical notes

Chapter 2 Kronecker products, vec operator, and Moore-Penrose inverse

2 The Kronecker product

3 Eigenvalues of a Kronecker product

4 The vec operator

5 The Moore-Penrose (MP) inverse

6 Existence and uniqueness of the MP inverse

7 Some properties of the MP inverse

8 Further properties

9 The solution of linear equation systems

Chapter 3 Miscellaneous matrix results

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 X1A + X2B′ = G1,X1B = G2

Bibliographical notes.

Part Two - Differentials: the theory

Chapter 4 Mathematical preliminaries

2 Interior points and accumulation points

3 Open and closed sets

4 The Bolzano-Weierstrass theorem

5 Functions

6 The limit of a function

7 Continuous functions and compactness

8 Convex sets

9 Convex and concave functions

Chapter 5 Differentials and differentiability

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

Chapter 6 The second differential

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

Chapter 7 Static optimization

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

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

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

2 The second identification table

3 Linear and quadratic forms

4 A useful theorem

5 The determinant function

6 The eigenvalue function

7 Other examples

8 Composite functions

9 The eigenvector function

10 Hessian of matrix functions, I

11 Hessian of matrix functions, II

Part Four - Inequalities

Chapter 11 Inequalities

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 λ1 and convexity of λn

7 Variational description of eigenvalues

8 Fischer's min-max theorem

9 Monotonicity of the eigenvalues

10 The Poincar´e separation theorem

11 Two corollaries of Poincar´e's theorem

12 Further consequences of the Poincar´e 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 (Σapi)1/p

20 A representation theorem for (trAp)1/p

21 H¨older's inequality

22 Concavity of log |A|

23 Minkowski's inequality

24 Quasilinear representation of |A|1/n

25 Minkowski's determinant theorem

26 Weighted means of order p

27 Schl¨omilch'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

Part Five - The linear model.

Chapter 12 Statistical preliminaries

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

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 case M(R′) ⊂M(X′)

9 Linear constraints: the general case

10 Linear constraints: the case M(R′) ∩M(X′) = {0}

11 A singular variance matrix: the case M(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

2 Best quadratic unbiased estimation of σ2

3 The best quadratic and positive unbiased estimator of σ2

4 The best quadratic unbiased estimator of σ2

5 Best quadratic invariant estimation of σ2

6 The best quadratic and positive invariant estimator of σ2

7 The best quadratic invariant estimator of σ2

8 Best quadratic unbiased estimation: multivariate normal case

9 Bounds for the bias of the least-squares estimator of σ2, I

10 Bounds for the bias of the least-squares estimator of σ2, II.

11 The prediction of disturbances.

Notes:

Includes bibliographical references (p. 449-465) and indexes

Includes bibliographical references and index.

Description based on print version record.

ISBN:

1-119-54121-2

1-119-54116-6

1-119-54119-0

OCLC:

1081339087