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MATRIX ANALYSIS
FOR STATISTICS
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MATRIX ANALYSIS
FOR STATISTICS
Third Edition
JAMESR.SCHOTT
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LibraryofCongressCataloging-in-PublicationData:
Names:Schott,JamesR.,1955-author.
Title:Matrixanalysisforstatistics/JamesR.Schott.
Description:Thirdedition.|Hoboken,NewJersey:JohnWiley&Sons,2016.
|Includesbibliographicalreferencesandindex.
Identifiers:LCCN2016000005|ISBN9781119092483(cloth)|ISBN9781119092469
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Subjects:LCSH:Matrices.|Mathematicalstatistics.
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ToSusan,Adam,andSarah
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CONTENTS
Preface xi
AbouttheCompanionWebsite xv
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1 AReviewofElementaryMatrixAlgebra 1
1.1 Introduction, 1
1.2 DefinitionsandNotation, 1
1.3 MatrixAdditionandMultiplication, 2
1.4 TheTranspose, 3
1.5 TheTrace, 4
1.6 TheDeterminant, 5
1.7 TheInverse, 9
1.8 PartitionedMatrices, 12
1.9 TheRankofaMatrix, 14
1.10 OrthogonalMatrices, 15
1.11 QuadraticForms, 16
1.12 ComplexMatrices, 18
1.13 RandomVectorsandSomeRelatedStatisticalConcepts, 19
Problems, 29
2 VectorSpaces 35
2.1 Introduction, 35
2.2 Definitions, 35
2.3 LinearIndependenceandDependence, 42
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viii CONTENTS
2.4 MatrixRankandLinearIndependence, 45
2.5 BasesandDimension, 49
2.6 OrthonormalBasesandProjections, 53
2.7 ProjectionMatrices, 58
2.8 LinearTransformationsandSystemsofLinearEquations, 65
2.9 TheIntersectionandSumofVectorSpaces, 73
2.10 ObliqueProjections, 76
2.11 ConvexSets, 80
Problems, 85
3 EigenvaluesandEigenvectors 95
3.1 Introduction, 95
3.2 Eigenvalues,Eigenvectors,andEigenspaces, 95
3.3 SomeBasicPropertiesofEigenvaluesandEigenvectors, 99
3.4 SymmetricMatrices, 106
3.5 ContinuityofEigenvaluesandEigenprojections, 114
3.6 ExtremalPropertiesofEigenvalues, 116
3.7 AdditionalResultsConcerningEigenvaluesOfSymmetric
Matrices, 123
3.8 NonnegativeDefiniteMatrices, 129
3.9 AntieigenvaluesandAntieigenvectors, 141
(cid:2) Problems, 144 (cid:2)
4 MatrixFactorizationsandMatrixNorms 155
4.1 Introduction, 155
4.2 TheSingularValueDecomposition, 155
4.3 TheSpectralDecompositionofaSymmetricMatrix, 162
4.4 TheDiagonalizationofaSquareMatrix, 169
4.5 TheJordanDecomposition, 173
4.6 TheSchurDecomposition, 175
4.7 TheSimultaneousDiagonalizationofTwoSymmetricMatrices, 178
4.8 MatrixNorms, 184
Problems, 191
5 GeneralizedInverses 201
5.1 Introduction, 201
5.2 TheMoore–PenroseGeneralizedInverse, 202
5.3 SomeBasicPropertiesoftheMoore–PenroseInverse, 205
5.4 TheMoore–PenroseInverseofaMatrixProduct, 211
5.5 TheMoore–PenroseInverseofPartitionedMatrices, 215
5.6 TheMoore–PenroseInverseofaSum, 219
5.7 TheContinuityoftheMoore–PenroseInverse, 222
5.8 SomeOtherGeneralizedInverses, 224
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Description:An up-to-date version of the complete, self-contained introduction to matrix analysis theory and practice Providing accessible and in-depth coverage of the most common matrix methods now used in statistical applications, Matrix Analysis for Statistics, Third Edition features an easy-to-follow theore
