Matrices: Theory and Applications

291 Pages·2010·3.906 MB·English

by  Denis Serre

Graduate Texts in Mathematics 216
Editorial Board
S. Axler
K.A. Ribet
For other titles in this series, go to
http://www.springer.com/series/136
Denis Serre
Matrices
Theory and Applications
Second Edition
1 C
Denis Serre
Unité de Mathématiques Pures et Appliquées
École Normale Supérieure de Lyon
69364 Lyon Cedex 07
France
[email protected]
Editorial Board:
S. Axler K. A. Ribet
Mathematics Department Mathematics Department
San Francisco State University University of California at Berkeley
San Francisco, CA 94132 Berkeley, CA 94720
USA USA
[email protected] [email protected]

ISSN 0072-5285
ISBN 978-1-4419-7682-6 e-ISBN 978-1-4419-7683-3
DOI 10.1007/978-1-4419-7683-3
Springer New York Dordrecht Heidelberg London
Library of Congress Control Number: 2010938321
Mathematics Subject Classification (2010): 15-XX, 22-XX, 47-XX, 65-XX
© Springer Science+Business Media, LLC 2010
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ToPascale,Fanny,Paul,andJoachim
Contents
PrefacefortheSecondEdition ....................................... xi
PrefacefortheFirstEdition ......................................... xiii
1 ElementaryLinearandMultilinearAlgebra ...................... 1
1.1 VectorsandScalars ......................................... 1
1.2 LinearMaps............................................... 5
1.3 BilinearMaps ............................................. 9
2 WhatAreMatrices............................................. 15
2.1 Introduction ............................................... 15
2.2 MatricesasLinearMaps..................................... 19
2.3 MatricesandBilinearForms ................................. 28
3 SquareMatrices ............................................... 31
3.1 Determinant ............................................... 31
3.2 Minors ................................................... 34
3.3 Invertibility ............................................... 38
3.4 EigenvaluesandEigenvectors ................................ 42
3.5 TheCharacteristicPolynomial................................ 43
3.6 Diagonalization ............................................ 48
3.7 Trigonalization............................................. 49
3.8 Rank-OnePerturbations ..................................... 52
3.9 AlternateMatricesandthePfaffian............................ 54
3.10 CalculatingtheCharacteristicPolynomial ...................... 56
3.11 IrreducibleMatrices ........................................ 59
4 TensorandExteriorProducts ................................... 69
4.1 TensorProductofVectorSpaces.............................. 69
4.2 ExteriorCalculus........................................... 72
4.3 TensorizationofLinearMaps ................................ 77
4.4 APolynomialIdentityinM (K).............................. 78
n
vii
viii Contents
5 MatriceswithRealorComplexEntries........................... 83
5.1 SpecialMatrices ........................................... 83
5.2 EigenvaluesofReal-andComplex-ValuedMatrices.............. 86
5.3 SpectralDecompositionofNormalMatrices.................... 91
5.4 NormalandSymmetricReal-ValuedMatrices................... 92
5.5 FunctionalCalculus ........................................ 94
5.6 NumericalRange........................................... 98
5.7 TheGershgorinDomain.....................................102
6 HermitianMatrices ............................................109
6.1 TheSquareRootoverHPD .................................109
n
6.2 RayleighQuotients .........................................110
6.3 FurtherPropertiesoftheSquareRoot..........................114
6.4 SpectrumofRestrictions ....................................115
6.5 SpectrumversusDiagonal ...................................117
6.6 TheDeterminantofNonnegativeHermitianMatrices.............119
7 Norms ........................................................127
7.1 ABriefReview ............................................127
7.2 Householder’sTheorem .....................................133
7.3 AnInterpolationInequality ..................................135
7.4 VonNeumann’sInequality...................................137
8 NonnegativeMatrices ..........................................149
8.1 NonnegativeVectorsandMatrices ............................149
8.2 ThePerron–FrobeniusTheorem:WeakForm ...................150
8.3 ThePerron–FrobeniusTheorem:StrongForm ..................151
8.4 CyclicMatrices ............................................154
8.5 StochasticMatrices.........................................156
9 MatriceswithEntriesinaPrincipalIdealDomain;JordanReduction163
9.1 Rings,PrincipalIdealDomains ...............................163
9.2 InvariantFactorsofaMatrix .................................167
9.3 SimilarityInvariantsandJordanReduction .....................170
10 ExponentialofaMatrix,PolarDecomposition,andClassicalGroups 183
10.1 ThePolarDecomposition....................................183
10.2 ExponentialofaMatrix .....................................184
10.3 StructureofClassicalGroups.................................188
10.4 TheGroupsU(p,q).........................................191
10.5 TheOrthogonalGroupsO(p,q) ..............................192
10.6 TheSymplecticGroupSp ..................................195
n
Contents ix
11 MatrixFactorizationsandTheirApplications .....................207
11.1 TheLU Factorization .......................................208
11.2 CholeskiFactorization ......................................213
11.3 TheQRFactorization .......................................214
11.4 SingularValueDecomposition................................216
11.5 TheMoore–PenroseGeneralizedInverse.......................218
12 IterativeMethodsforLinearSystems ............................225
12.1 AConvergenceCriterion ....................................226
12.2 BasicMethods.............................................227
12.3 TwoCasesofConvergence ..................................229
12.4 TheTridiagonalCase .......................................231
12.5 TheMethodoftheConjugateGradient.........................235
13 ApproximationofEigenvalues ...................................247
13.1 GeneralConsiderations......................................247
13.2 HessenbergMatrices........................................249
13.3 TheQRMethod............................................253
13.4 TheJacobiMethod .........................................259
13.5 ThePowerMethods ........................................266
References.........................................................277
IndexofNotations..................................................279
GeneralIndex .....................................................283
CitedNames .......................................................289
Preface for the Second Edition
Itisonlyafterlongusethatanauthorrealizestheflawsandthegapsinhisorher
book. Having taught from a significant part of it, having gathered more than three
hundredexercisesonapublicwebsitehttp://www.umpa.ens-lyon.fr/
serre/DPF/exobis.pdf,andhavinglearnedalotwithineightyearsofread-
ing after the first edition was published, I arrived at the conclusion that a second
editionofMatrices:TheoryandApplicationsshouldbesignificantlydifferentfrom
thefirstone.
First of all, I felt ashamed of the very light presentation of the backgrounds in
linear algebra and elementary matrix theory in Chapter 1. In French, I should say
cen’e´taitnifait,nia` faire(neitherdone,nortobedone).Ithusbeganbyrewriting
this part completely, taking this opportunity to split pure linear algebra from the
introductiontomatrices.IhopethatthereaderissatisfiedwiththenewChapters1
and2below.
When teaching, it was easy to recognize the lack of logical structure here and
there.Forthesakeofamoreelegantpresentation,Ithereforemovedseveralstate-
ments from one chapter to another. It even happened that entire sections was dis-
placed,suchasthoseaboutsingularvaluedecomposition,theLeverrieralgorithm,
theSchurcomplement,orthesquarerootofpositiveHermitianmatrices.
Next, I realized that some important material was missing in the first edition.
Thishasledmetoincreasethesizeofthisbookbyaboutfortypercent.Thenewly
addedtopicsare
• Dunforddecomposition
• Calculuswithrank-oneperturbations
• ImprovementbyPreparataandSarwateofLeverrier’salgorithmforcalculating
thecharacteristicpolynomial
• Tensorcalculus
• PolynomialidentityofAmitsurandLevitzki
• Regularityofsimpleeigenvaluesforcomplexmatrices
• FunctionalcalculusandtheDunford–Taylorformula
• Stableandunstablesubspaces
xi
xii PrefacefortheSecondEdition
• Numericalrange
• Weylinequalities
• Concavityof(detH)1/noverHPD
n
• vonNeumanninequality
• ConvergenceoftheJacobimethodwithrandomchoice(perhapsanewresult).
Withsomanyadditions,thechapteronrealandcomplexmatricesextendedbeyond
areasonablesize.Thisinturnledmetosplitit,bydedicatingaspecificchapterto
thestudyofHermitianmatrices.Becausetensorcalculus,togetherwithpolynomial
identities,alsoformsanewchapter,thenumberofchaptershasincreasedfromten
tothirteen.
The reader might wonder why I included the new fourth chapter, because it is
essentially not used in the sequel. Several reasons led me to this choice. First of
all,tensorandexteriorcalculusarefundamentalinspectralanalysis,indifferential
geometry, in theoretical physics and many other domains. Next, I think that the
theoremofAmitsurandLevitzkiisoneofthemostbeautifulinmatrixtheory,which
remained mysterious until a quite direct proof using exterior algebra was found,
allowing it to be presented in a graduate textbook. The presence of this result
isconsistentwithmyphilosophythateverychapter,beyondtheintroductoryones,
shouldcontainatleastoneadvancedstatement.Mylastmotivationwastoinclude
morealgebraicissues,becauseitisapermanenttendancyofmatrixanalysistobe
tooanalytic.
Followingthispointofview,IshouldhavelovedtoincludeaproofofA.Horn’s
conjecture, after W. Fulton, A. Klyachko, A. Knutson and T. Tao: the statement,
whichhasananalyticflavorinasmuchasitconsistsininequalitiesbetweeneigen-
values of Hermitian matrices, is undoubtedly a great achievement of algebraists,
involving,asitdoes,representationtheoryandSchubertcalculus.However,thema-
terialtobeincludedwouldhavebeenfartooadvanced.Thistheoryisstilltoofresh
and will not be part of textbooks before it has been digested, and this will take
perhapsafewdecades.ThusIdecidedtobecontentwithdiscussingtheWeyland
Lidski˘ıinequalities,whicharethelowerstepsinHorn’slist.
Eventhoughthisnewversionhadexpandedalot,Ithoughtlongandhardabout
including a fundamental and beautiful (to my opinion) issue, namely Loewner’s
theory of operator monotone functions. I eventually abandoned this idea because
the topic would have needed too much space, thus upsetting the balance between
elementaryandadvancedmaterial.
Finally,Ihaveincludedmanynewexercises,manyofthemofferingawaytogo
furtherintothetheoryandpractice.Thewebsitementionedaboveremainsavailable,
andis expectedtogrow overtime,but thereferencesin itwillremain unchanged,
in order that it may be usable by the owners of the first edition, until I feel that it
absolutelyneedstoberefreshed.
Lyon,France DenisSerre
November2009
Preface for the First Edition
The study of matrices occupies a singular place within mathematics. It is still an
areaofactiveresearch,anditisusedbyeverymathematicianandbymanyscientists
workinginvariousspecialities.Severalexamplesillustrateitsversatility:
• Scientificcomputinglibrariesbegangrowingaroundmatrixcalculus.Asamatter
offact,thediscretizationofpartialdifferentialoperatorsisanendlesssourceof
linearfinite-dimensionalproblems.
• Atadiscretelevel,themaximumprincipleisrelatedtononnegativematrices.
• Controltheoryandstabilizationofsystemswithfinitelymanydegreesoffreedom
involvespectralanalysisofmatrices.
• ThediscreteFouriertransform,includingthefastFouriertransform,makesuse
ofToeplitzmatrices.
• Statisticsiswidelybasedoncorrelationmatrices.
• Thegeneralizedinverseisinvolvedinleast-squaresapproximation.
• Symmetric matrices are inertia, deformation, or viscous tensors in continuum
mechanics.
• Markovprocessesinvolvestochasticorbistochasticmatrices.
• Graphscanbedescribedinausefulwaybysquarematrices.
• Quantum chemistry is intimately related to matrix groups and their representa-
tions.
• The case of quantum mechanics is especially interesting. Observables are Her-
mitianoperators;theireigenvaluesareenergylevels.Intheearlyyears,quantum
mechanicswascalled“mechanicsofmatrices,”andithasnowgivenrisetothe
development of the theory of large random matrices. See [25] for a thorough
accountofthisfashionabletopic.
Thistextwasconceivedduringtheyears1998–2001,ontheoccasionofacourse
that I taught at the E´cole Normale Supe´rieure de Lyon. As such, every result is
accompanied by a detailed proof. During this course I tried to investigate all the
principalmathematicalaspectsofmatrices:algebraic,geometric,andanalytic.
In some sense, this is not a specialized book. For instance, it is not as detailed
as [19] concerning numerics, or as [40] on eigenvalue problems, or as [21] about
xiii