Matrix Differential Calculus With Applications in Statistics and Econometrics

by Magnus, Jan R.; Neudecker, Heinz

Edition: Revised

ISBN13: 9780471986324

ISBN10: 0471986321

Format: Hardcover

Pub. Date: 1998-12-01

Publisher(s): John Wiley & Sons Inc

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Summary

Matrix Differential Calculus With Applications in Statistics and Econometrics Revised Edition Jan R. Magnus, CentER, Tilburg University, The Netherlands and Heinz Neudecker, Cesaro, Schagen, The Netherlands " .deals rigorously with many of the problems that have bedevilled the subject up to the present time." - Stephen Pollock, Econometric Theory "I continued to be pleasantly surprised by the variety and usefulness of its contents " - Isabella Verdinelli, Journal of the American Statistical Association Continuing the success of their first edition, Magnus and Neudecker present an exhaustive and self-contained revised text on matrix theory and matrix differential calculus. Matrix calculus has become an essential tool for quantitative methods in a large number of applications, ranging from social and behavioural sciences to econometrics. While the structure and successful elements of the first edition remain, this revised and updated edition contains many new examples and exercises. * Contains the essentials of multivariable calculus with an emphasis on the use of differentials * Many new examples and exercises * Fulfils the need for a unified and self-contained treatment of matrix differential calculus * Includes new developments in this field Part I presents a concise, yet thorough overview of matrix algebra, while the second part develops the theory of differentials. The remaining Parts III to VI combine the theory and application of matrix differential calculus providing the practitioner and researcher with both a quick review and a detailed reference. Visit our web page http://www.wiley.com/

Preface

Preface to the first revised printing

xvii

Preface to the second revised printing

xviii

Part One---Matrices

Basic properties of vectors and matrices

(24)

Introduction

(1)

Sets

(1)

Matrices: addition and multiplication

(1)

The transpose of a matrix

(1)

Square matrices

(1)

Linear forms and quadratic forms

(1)

The rank of a matrix

(1)

The inverse

(1)

The determinant

(1)

The trace

(1)

Partitioned matrices

(2)

Complex matrices

(1)

Eigenvalues and eigenvectors

(3)

Schur's decomposition theorem

(1)

The Jordan decomposition

(1)

The singular-value decomposition

(1)

Further results concerning eigenvalues

(2)

Positive (semi) definite matrices

(2)

Three further results for positive definite matrices

(1)

A useful result

(3)

Miscellaneous exercises

(1)

Bibliographical notes

(1)

Kronecker products, the vec operator and the Moore-Penrose inverse

(13)

Introduction

(1)

The Kronecker product

(1)

Eigenvalues of a Kronecker product

(2)

The vec operator

(2)

The Moore-Penrose (MP) inverse

(1)

Existence and uniqueness of the MP inverse

(1)

Some properties of the MP inverse

(1)

Further properties

(2)

The solution of linear equation systems

(4)

Miscellaneous exercises

(1)

Bibliographical notes

(1)

Miscellaneous matrix results

(25)

Introduction

(1)

The adjoint matrix

(1)

Proof of Theorem 1

(2)

Two results concerning bordered determinants

(1)

The matrix equation AX = 0

(1)

The Hadamard product

(1)

The commutation matrix Kmn

(2)

The duplication matrix Dn

(2)

Relationship between Dn + 1 and Dn, I

(2)

Relationship between Dn + 1 and Dn, II

(1)

Conditions for a quadratic form to be positive (negative) subject to linear constraints

(3)

Necessary and sufficient conditions for r(A:B) = r(A) + r(B)

(1)

The bordered Gramian matrix

(3)

The equations X1A + X2B = G1, X1B = G2

(5)

Miscellaneous exercises

(1)

Bibliographical notes

(3)

Part Two---Differentials: the theory

Mathematical preliminaries

(13)

Introduction

(1)

Interior points and accumulation points

(1)

Open and closed sets

(3)

The Bolzano-Weierstrass theorem

(1)

Functions

(1)

The limit of a function

(1)

Continuous functions and compactness

(1)

Convex sets

(3)

Convex and concave functions

(3)

Bibliographical notes

(1)

Differentials and differentiability

(21)

Introduction

(1)

Continuity

(2)

Differentiability and linear approximation

(2)

The differential of a vector function

(2)

Uniqueness of the differential

(1)

Continuity of differentiable functions

(1)

Partial derivatives

(2)

The first identification theorem

(1)

Existence of the differential, I

(1)

Existence of the differential, II

(2)

Continuous differentiability

(1)

The chain rule

(2)

Cauchy invariance

(1)

The mean-value theorem for real-valued functions

(1)

Matrix functions

(2)

Some remarks on notation

(3)

Miscellaneous exercises

(1)

Bibliographical notes

(1)

The second differential

(17)

Introduction

(1)

Second-order partial derivatives

(1)

The Hessian matrix

100

(1)

Twice differentiability and second-order approximation, I

101

(1)

Definition of twice differentiability

102

(1)

The second differential

103

(2)

(Column) symmetry of the Hessian matrix

105

(2)

The second identification theorem

107

(1)

Twice differentiability and second-order approximation, II

108

(2)

Chain rule for Hessian matrices

110

(1)

The analogue for second differentials

111

(1)

Taylor's theorem for real-valued functions

112

(1)

Higher-order differentials

113

(1)

Matrix functions

114

(2)

Bibliographical notes

115

(1)

Static optimization

116

(31)

Introduction

116

(1)

Unconstrained optimization

116

(2)

The existence of absolute extrema

118

(1)

Necessary conditions for a local minimum

119

(2)

Sufficient conditions for a local minimum: first-derivative test

121

(1)

Sufficient conditions for a local minimum: second-derivative test

122

(2)

Characterization of differentiable convex functions

124

(3)

Characterization of twice differentiable convex functions

127

(1)

Sufficient conditions for an absolute minimum

128

(1)

Monotonic transformations

129

(1)

Optimization subject to constraints

130

(1)

Necessary conditions for a local minimum under constraints

131

(4)

Sufficient conditions for a local minimum under constraints

135

(4)

Sufficient conditions for an absolute minimum under constraints

139

(1)

A note on constraints in matrix form

140

(1)

Economic interpretation of Lagrange multipliers

141

(6)

Appendix: the implicit function theorem

142

(2)

Bibliographical notes

144

(3)

Part Three---Differentials: the practice

Some important differentials

147

(23)

Introduction

147

(1)

Fundamental rules of differential calculus

147

(2)

The differential of a determinant

149

(2)

The differential of an inverse

151

(1)

The differential of the Moore-Penrose inverse

152

(3)

The differential of the adjoint matrix

155

(2)

On differentiating eigenvalues and eigenvectors

157

(1)

The differential of eigenvalues and eigenvectors: the real symmetric case

158

(3)

The differential of eigenvalues and eigenvectors: the general complex case

161

(2)

Two alternative expressions for dλ

163

(3)

The second differential of the eigenvalue function

166

(1)

Multiple eigenvalues

167

(3)

Miscellaneous exercises

167

(2)

Bibliographical notes

169

(1)

First-order differentials and Jacobian matrices

170

(18)

Introduction

170

(1)

Classification

170

(1)

Bad notation

171

(2)

Good notation

173

(1)

Identification of Jacobian matrices

174

(1)

The first identification table

175

(1)

Partitioning of the derivative

175

(1)

Scalar functions of a vector

176

(1)

Scalar functions of a matrix, I: trace

177

(1)

Scalar functions of a matrix, II: determinant

178

(2)

Scalar functions of a matrix, III: eigenvalue

180

(1)

Two examples of vector functions

181

(1)

Matrix functions

182

(2)

Kronecker products

184

(1)

Some other problems

185

(3)

Bibliographical notes

187

(1)

Second-order differentials and Hessian matrices

188

(11)

Introduction

188

(1)

The Hessian matrix of a matrix function

188

(1)

Identification of Hessian matrices

189

(1)

The second identification table

190

(1)

An explicit formula for the Hessian matrix

191

(1)

Scalar functions

192

(2)

Vector functions

194

(1)

Matrix functions, I

194

(1)

Matrix functions, II

195

(4)

Part Four---Inequalities

Inequalities

199

(44)

Introduction

199

(1)

The Cauchy-Schwarz inequality

199

(2)

Matrix analogues of the Cauchy-Schwarz inequality

201

(1)

The theorem of the arithmetic and geometric means

202

(1)

The Rayleigh quotient

203

(1)

Concavity of λ1, convexity of λn

204

(1)

Variational description of eigenvalues

205

(1)

Fischer's min-max theorem

206

(2)

Monotonicity of the eigenvalues

208

(1)

The Poincar'e separation theorem

209

(1)

Two corollaries of Poincare's theorem

210

(1)

Further consequences of the Poincare theorem

211

(1)

Multiplicative version

212

(1)

The maximum of a bilinear form

213

(1)

Hadamard's inequality

214

(1)

An interlude: Karamata's inequality

215

(2)

Karamata's inequality applied to eigenvalues

217

(1)

An inequality concerning positive semidefinite matrices

217

(1)

A representation theorem for (Σap)1/p

218

(1)

A representation theorem for (tr Ap)1/p

219

(1)

Holder's inequality

220

(2)

Concavity of log |A|

222

(1)

Minkowski's inequality

223

(1)

Quasilinear representation of |A|1/n

224

(3)

Minkowski's determinant theorem

227

(1)

Weighted means of order p

227

(2)

Schlomilch's inequality

229

(1)

Curvature properties of Mp(x, a)

230

(2)

Least squares

232

(1)

Generalized least squares

233

(1)

Restricted least squares

233

(2)

Restricted least squares: matrix version

235

(8)

Miscellaneous exercises

236

(4)

Bibliographical notes

240

(3)

Part Five---The linear model

Statistical preliminaries

243

(11)

Introduction

243

(1)

The cumulative distribution function

243

(1)

The joint density function

244

(1)

Expectations

244

(1)

Variance and covariance

245

(2)

Independence of two random variables (vectors)

247

(2)

Independence of n random variables (vectors)

249

(1)

Sampling

249

(1)

The one-dimensional normal distribution

249

(1)

The multivariate normal distribution

250

(2)

Estimation

252

(2)

Miscellaneous exercises

253

(1)

Bibliographical notes

253

(1)

The linear regression model

254

(33)

Introduction

254

(1)

Affine minimum-trace unbiased estimation

255

(1)

The Gauss-Markov theorem

256

(2)

The method of least squares

258

(1)

Aitken's theorem

259

(2)

Multicollinearity

261

(2)

Estimable functions

263

(1)

Linear constraints: the case M(R') ⊂ M (X')

264

(3)

Linear constraints: the general case

267

(3)

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

270

(1)

A singular variance matrix: the case M (X) ⊂ M (V)

271

(2)

A singular variance matrix: the case r(X' V+ X) = r(X)

273

(1)

A singular variance matrix: the general case, I

274

(1)

Explicit and implicit linear constraints

275

(2)

The general linear model, I

277

(1)

A singular variance matrix: the general case, II

278

(3)

The general linear model, II

281

(1)

Generalized least squares

282

(1)

Restricted least squares

283

(4)

Miscellaneous exercises

285

(1)

Bibliographical notes

286

(1)

Further topics in the linear model

287

(26)

Introduction

287

(1)

Best quadratic unbiased estimation of σ2

287

(1)

The best quadratic and positive unbiased estimator of 7sigma;2

288

(2)

The best quadratic unbiased estimator of σ2

290

(2)

Best quadratic invariant estimation of σ2

292

(1)

The best quadratic and positive invariant estimator of σ2

293

(1)

The best quadratic invariant estimator of σ2

294

(1)

Best quadratic unbiased estimation in the multivariate normal case

295

(2)

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

297

(2)

Bounds for the bias of the least squares estimator of σ2, II

299

(1)

The prediction of disturbances

300

(1)

Predictors that are best linear unbiased with scalar variance matrix (Blus)

301

(2)

Predictors that are best linear unbiased with fixed variance matrix (Bluf), I

303

(2)

Predictors that are best linear unbiased with fixed variance matrix (Bluf), II

305

(1)

Local sensitivity of the posterior mean

306

(2)

Local sensitivity of the posterior precision

308

(5)

Bibliographical notes

309

(4)

Part Six---Applications to maximum likelihood estimation

Maximum likelihood estimation

313

(18)

Introduction

313

(1)

The method of maximum likelihood (M L)

313

(1)

M L estimation of the multivariate normal distribution

314

(2)

Implicit versus explicit treatment of symmetry

316

(1)

The treatment of positive definiteness

317

(1)

The information matrix

317

(2)

M L estimation of the multivariate normal distribution with distinct means

319

(1)

The multivariate linear regression model

320

(2)

The errors-in-variables model

322

(2)

The nonlinear regression model with normal errors

324

(2)

A special case: functional independence of mean parameters and variance parameters

326

(1)

Generalization of Theorem 6

327

(4)

Miscellaneous exercises

329

(1)

Bibliographical notes

330

(1)

Simultaneous equations

331

(21)

Introduction

331

(1)

The simultaneous equations model

331

(2)

The identification problem

333

(1)

Identification with linear constraints on B and τ only

334

(1)

Identification with linear constraints on B, τ and Σ

335

(2)

Nonlinear constraints

337

(1)

Full-information maximum likelihood (Fiml): the information matrix (general case)

337

(2)

Full-information maximum likelihood (Fiml): the asymptotic variance matrix (special case)

339

(3)

Limited-information maximum likelihood (Liml): the first-order conditions

342

(2)

Limited-information maximum likelihood (Liml): the information matrix

344

(2)

Limited-information maximum likelihood (Liml): the asymptotic variance matrix

346

(6)

Bibliographical notes

351

(1)

Topics in psychometrics

352

(27)

Introduction

352

(1)

Population principal components

353

(1)

Optimality of principal components

353

(2)

A related result

355

(1)

Sample principal components

356

(2)

Optimality of sample principal components

358

(1)

Sample analogue of Theorem 3

358

(1)

One-mode component analysis

358

(3)

Relationship between one-mode component analysis and sample principal components

361

(1)

Two-mode component analysis

362

(1)

Multimode component analysis

363

(3)

Factor analysis

366

(3)

A zigzag routine

369

(1)

A Newton-Raphson routine

370

(3)

Kaiser's varimax method

373

(3)

Canonical correlations and variates in the population

376

(3)

Bibliographical notes

378

(1)

Bibliography

379

(8)

Index of Symbols

387

(3)

Subject Index

390

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