Matrix Algebra

466 Pages·2005·1.766 MB·English

by  Abadir K.M., Magnus J.R.

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EconometricExercises,Volume1
Matrix Algebra
Matrix Algebra is the first volume of the Econometric Exercises Series. It contains ex-
ercises relating to course material in matrix algebra that students are expected to know
while enrolled in an (advanced) undergraduate or a postgraduate course in econometrics
or statistics. The book contains a comprehensive collection of exercises, all with full
answers. Butthe bookis notjusta collectionof exercises; infact, itisatextbook, though
onethatisorganizedinacompletelydifferentmannerthantheusualtextbook. Thevolume
canbeusedeitherasaself-containedcourseinmatrixalgebraorasasupplementarytext.
Karim Abadir has held a joint Chair since 1996 in the Departments of Mathematics and
EconomicsattheUniversityofYork,wherehehasbeenthefounderanddirectorofvarious
degree programs. He has also taught at the American University in Cairo, the University
of Oxford, and the University of Exeter. He became an Extramural Fellow at CentER
(Tilburg University) in 2003. Professor Abadir is a holder of two Econometric Theory
Awards, and has authored many articles intop journals, including the Annals ofStatistics,
Econometric Theory, Econometrica, and the Journal of Physics A. He is Coordinating
Editor (and one of the founding editors) of the Econometrics Journal, and Associate
Editor of Econometric Reviews, Econometric Theory, Journal of Financial Econometrics,
andPortugueseEconomicJournal.
Jan Magnus is Professor of Econometrics, CentER and Department of Econometrics and
Operations Research, Tilburg University, the Netherlands. He has also taught at the
University of Amsterdam, The University of British Columbia, The London School of
Economics, The University of Montreal, and The European University Institute among
other places. His books include Matrix Differential Calculus (with H. Neudecker), Linear
Structures,MethodologyandTacitKnowledge(withM.S.Morgan),andEconometrics: A
FirstCourse(inRussianwithP.K.KatyshevandA.A.Peresetsky). ProfessorMagnushas
written numerous articles in the leading journals, including Econometrica, The Annals of
Statistics, The Journal of the American Statistical Association, Journal of Econometrics,
LinearAlgebraandItsApplications,andTheReviewofIncomeandWealth. HeisaFellow
oftheJournalofEconometrics,holderoftheEconometricTheoryAward,andassociateed-
itorofTheJournalofEconomicMethodology,ComputationalStatisticsandDataAnalysis,
andtheJournalofMultivariateAnalysis.
EconometricExercises
Editors:
KarimM.Abadir,DepartmentsofMathematicsandEconomics,
UniversityofYork,UK
JanR.Magnus,CentERandDepartmentofEconometricsandOperationsResearch,
TilburgUniversity,TheNetherlands
PeterC.B.Phillips,CowlesFoundationforResearchinEconomics,
YaleUniversity,USA
TitlesintheSeries(*=planned):
1 MatrixAlgebra(K.M.AbadirandJ.R.Magnus)
2 Statistics(K.M.Abadir,R.D.H.HeijmansandJ.R.Magnus)
3 EconometricModels,I:Theory(P.Paruolo)
4 EconometricModels,I:EmpiricalApplications(A.vanSoestandM.Verbeek)
* EconometricModels,II:Theory
* EconometricModels,II:EmpiricalApplications
* TimeSeriesEconometrics,I
* TimeSeriesEconometrics,II
* Microeconometrics
* PanelData
* BayesianEconometrics
* NonlinearModels
* NonparametricsandSemiparametrics
* Simulation-BasedEconometricMethods
* ComputationalMethods
* FinancialEconometrics
* Robustness
* EconometricMethodology
Matrix Algebra
KarimM.Abadir
DepartmentsofMathematicsandEconomics,UniversityofYork,UK
JanR.Magnus
CentERandDepartmentofEconometricsandOperationsResearch,
TilburgUniversity,TheNetherlands
CAMBRIDGEUNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521822893
© Karim M. Abadir and Jan R. Magnus 2005
This publication is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format 2005
ISBN-13 978-0-511-34440-4 eBook (EBL)
ISBN-10 0-511-34440-6 eBook (EBL)
ISBN-13 978-0-521-82289-3 hardback
ISBN-10 0-521-82289-0 hardback
ISBN-13 978-0-521-53746-9 paperback
ISBN-10 0-521-53746-0 paperback
Cambridge University Press has no responsibility for the persistence or accuracy of urls
for external or third-party internet websites referred to in this publication, and does not
guarantee that any content on such websites is, or will remain, accurate or appropriate.
Tomyparents,andtoKouka,Ramez,Naguib,Ne´vine
ToGideonandHedda

Contents
Listofexercises xi
PrefacetotheSeries xxv
Preface xxix
1 Vectors 1
1.1 Realvectors 4
1.2 Complexvectors 11
2 Matrices 15
2.1 Realmatrices 19
2.2 Complexmatrices 39
3 Vector spaces 43
3.1 Complexandrealvectorspaces 47
3.2 Inner-productspace 61
3.3 Hilbertspace 67
4 Rank,inverse,anddeterminant 73
4.1 Rank 75
4.2 Inverse 83
4.3 Determinant 87
5 Partitionedmatrices 97
5.1 Basicresultsandmultiplicationrelations 98
5.2 Inverses 103
5.3 Determinants 109
5.4 Rank(in)equalities 119
5.5 Thesweepoperator 126
6 Systemsofequations 131
6.1 Elementarymatrices 132
6.2 Echelonmatrices 137
viii Contents
6.3 Gaussianelimination 143
6.4 Homogeneousequations 148
6.5 Nonhomogeneousequations 151
7 Eigenvalues,eigenvectors,andfactorizations 155
7.1 Eigenvaluesandeigenvectors 158
7.2 Symmetricmatrices 175
7.3 Someresultsfortriangularmatrices 182
7.4 Schur’sdecompositiontheoremanditsconsequences 187
7.5 Jordan’sdecompositiontheorem 192
7.6 Jordanchainsandgeneralizedeigenvectors 201
8 Positive (semi)definiteandidempotentmatrices 209
8.1 Positive(semi)definitematrices 211
8.2 Partitioningandpositive(semi)definitematrices 228
8.3 Idempotentmatrices 231
9 Matrixfunctions 243
9.1 Simplefunctions 246
9.2 Jordanrepresentation 255
9.3 Matrix-polynomialrepresentation 265
10 Kroneckerproduct,vec-operator,andMoore-Penroseinverse 273
10.1 TheKroneckerproduct 274
10.2 Thevec-operator 281
10.3 TheMoore-Penroseinverse 284
10.4 Linearvectorandmatrixequations 292
10.5 Thegeneralizedinverse 295
11 Patternedmatrices: commutation-andduplicationmatrix 299
11.1 Thecommutationmatrix 300
11.2 Thesymmetrizermatrix 307
11.3 Thevech-operatorandtheduplicationmatrix 311
11.4 Linearstructures 318
12 Matrixinequalities 321
12.1 Cauchy-Schwarztypeinequalities 322
12.2 Positive(semi)definitematrixinequalities 325
12.3 InequalitiesderivedfromtheSchurcomplement 341
12.4 Inequalitiesconcerningeigenvalues 343
13 Matrixcalculus 351
13.1 Basicpropertiesofdifferentials 355
13.2 Scalarfunctions 356
13.3 Vectorfunctions 360
13.4 Matrixfunctions 361
13.5 Theinverse 364
13.6 Exponentialandlogarithm 368
13.7 Thedeterminant 369