Matrix Analysis

662 Pages·2013·3.89 MB·English

by  Roger A. Horn

MatrixAnalysis
SecondEdition
Linear algebra and matrix theory are fundamental tools in mathematical and physical
science,aswellasfertilefieldsforresearch.Thisneweditionoftheacclaimedtextpresents
resultsofbothclassicandrecentmatrixanalysisusingcanonicalformsasaunifyingtheme,
anddemonstratestheirimportanceinavarietyofapplications.
The authors have thoroughly revised, updated, and expanded on the first edition. The
bookopenswithanextendedsummaryofusefulconceptsandfactsandincludesnumerous
newtopicsandfeatures,suchas:
(cid:2)
NewsectionsonthesingularvalueandCSdecompositions
(cid:2)
NewapplicationsoftheJordancanonicalform
(cid:2)
AnewsectionontheWeyrcanonicalform
(cid:2)
Expandedtreatmentsofinverseproblemsandofblockmatrices
(cid:2)
AcentralroleforthevonNeumanntracetheorem
(cid:2)
A new appendix with a modern list of canonical forms for a pair of Hermitian
matricesandforasymmetric–skewsymmetricpair
(cid:2)
Expandedindexwithmorethan3,500entriesforeasyreference
(cid:2)
Morethan1,100problemsandexercises,manywithhints,toreinforceunderstand-
inganddevelopauxiliarythemessuchasfinite-dimensionalquantumsystems,the
compoundandadjugatematrices,andtheLoewnerellipsoid
(cid:2)
Anewappendixprovidesacollectionofproblem-solvinghints.
RogerA.HornisaResearchProfessorintheDepartmentofMathematicsattheUniversity
ofUtah.HeistheauthorofTopicsinMatrixAnalysis(CambridgeUniversityPress1994).
CharlesR.JohnsonistheauthorofTopicsinMatrixAnalysis(CambridgeUniversityPress
1994).

Matrix Analysis
Second Edition
Roger A. Horn
UniversityofUtah
Charles R. Johnson
cambridgeuniversitypress
Cambridge,NewYork,Melbourne,Madrid,CapeTown,
Singapore,Sa˜oPaulo,Delhi,MexicoCity
CambridgeUniversityPress
32AvenueoftheAmericas,NewYork,NY10013-2473,USA
www.cambridge.org
Informationonthistitle:www.cambridge.org/9780521548236
(cid:2)C RogerA.HornandCharlesR.Johnson1985,2013
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished1985
Firstpaperbackedition1990
Secondeditionfirstpublished2013
PrintedintheUnitedStatesofAmerica
AcatalogrecordforthispublicationisavailablefromtheBritishLibrary.
LibraryofCongressCataloginginPublicationData
Horn,RogerA.
Matrixanalysis/RogerA.Horn,CharlesR.Johnson.–2nded.
p. cm.
Includesbibliographicalreferencesandindex.
ISBN978-0-521-83940-2(hardback)
1.Matrices. I.Johnson,CharlesR. II.Title.
QA188.H66 2012
512.9(cid:3)434–dc23 2012012300
ISBN978-0-521-83940-2Hardback
ISBN978-0-521-54823-6Paperback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyofURLsforexternal
orthird-partyInternetWebsitesreferredtointhispublicationanddoesnotguaranteethatanycontent
onsuchWebsitesis,orwillremain,accurateorappropriate.
Tothematrixtheorycommunity

Contents
PrefacetotheSecondEdition pagexi
PrefacetotheFirstEdition xv
0 ReviewandMiscellanea 1
0.0 Introduction 1
0.1 Vectorspaces 1
0.2 Matrices 5
0.3 Determinants 8
0.4 Rank 12
0.5 Nonsingularity 14
0.6 TheEuclideaninnerproductandnorm 15
0.7 Partitionedsetsandmatrices 16
0.8 Determinantsagain 21
0.9 Specialtypesofmatrices 30
0.10 Changeofbasis 39
0.11 Equivalencerelations 40
1 Eigenvalues,Eigenvectors,andSimilarity 43
1.0 Introduction 43
1.1 Theeigenvalue–eigenvectorequation 44
1.2 Thecharacteristicpolynomialandalgebraicmultiplicity 49
1.3 Similarity 57
1.4 Leftandrighteigenvectorsandgeometricmultiplicity 75
2 UnitarySimilarityandUnitaryEquivalence 83
2.0 Introduction 83
2.1 UnitarymatricesandtheQRfactorization 83
2.2 Unitarysimilarity 94
2.3 Unitaryandrealorthogonaltriangularizations 101
2.4 ConsequencesofSchur’striangularizationtheorem 108
2.5 Normalmatrices 131
vii
viii Contents
2.6 Unitaryequivalenceandthesingularvaluedecomposition 149
2.7 TheCSdecomposition 159
3 CanonicalFormsforSimilarityandTriangularFactorizations 163
3.0 Introduction 163
3.1 TheJordancanonicalformtheorem 164
3.2 ConsequencesoftheJordancanonicalform 175
3.3 Theminimalpolynomialandthecompanionmatrix 191
3.4 TherealJordanandWeyrcanonicalforms 201
3.5 Triangularfactorizationsandcanonicalforms 216
4 HermitianMatrices,SymmetricMatrices,andCongruences 225
4.0 Introduction 225
4.1 PropertiesandcharacterizationsofHermitianmatrices 227
4.2 Variationalcharacterizationsandsubspaceintersections 234
4.3 EigenvalueinequalitiesforHermitianmatrices 239
4.4 Unitarycongruenceandcomplexsymmetricmatrices 260
4.5 Congruencesanddiagonalizations 279
4.6 Consimilarityandcondiagonalization 300
5 NormsforVectorsandMatrices 313
5.0 Introduction 313
5.1 Definitionsofnormsandinnerproducts 314
5.2 Examplesofnormsandinnerproducts 320
5.3 Algebraicpropertiesofnorms 324
5.4 Analyticpropertiesofnorms 324
5.5 Dualityandgeometricpropertiesofnorms 335
5.6 Matrixnorms 340
5.7 Vectornormsonmatrices 371
5.8 Conditionnumbers:inversesandlinearsystems 381
6 LocationandPerturbationofEigenvalues 387
6.0 Introduction 387
6.1 Gersˇgorindiscs 387
6.2 Gersˇgorindiscs–acloserlook 396
6.3 Eigenvalueperturbationtheorems 405
6.4 Othereigenvalueinclusionsets 413
7 PositiveDefiniteandSemidefiniteMatrices 425
7.0 Introduction 425
7.1 Definitionsandproperties 429
7.2 Characterizationsandproperties 438
7.3 Thepolarandsingularvaluedecompositions 448
7.4 Consequencesofthepolarandsingularvaluedecompositions 458
7.5 TheSchurproducttheorem 477
7.6 Simultaneousdiagonalizations,products,andconvexity 485
7.7 TheLoewnerpartialorderandblockmatrices 493
7.8 Inequalitiesinvolvingpositivedefinitematrices 505
Contents ix
8 PositiveandNonnegativeMatrices 517
8.0 Introduction 517
8.1 Inequalitiesandgeneralities 519
8.2 Positivematrices 524
8.3 Nonnegativematrices 529
8.4 Irreduciblenonnegativematrices 533
8.5 Primitivematrices 540
8.6 Agenerallimittheorem 545
8.7 Stochasticanddoublystochasticmatrices 547
AppendixA ComplexNumbers 555
AppendixB ConvexSetsandFunctions 557
AppendixC TheFundamentalTheoremofAlgebra 561
AppendixD ContinuityofPolynomialZeroesandMatrix
Eigenvalues 563
AppendixE Continuity,Compactness,andWeierstrass’sTheorem 565
AppendixF CanonicalPairs 567
References 571
Notation 575
HintsforProblems 579
Index 607