Fundamentals of Matrix Analysis with Applications

409 Pages·2016·3.87 MB·English

by  Edward Barry Saff

FUNDAMENTALS OF MATRIX ANALYSIS
WITH APPLICATIONS

FUNDAMENTALS OF

MATRIX ANALYSIS

WITH APPLICATIONS

EDWARDBARRYSAFF

DepartmentofMathematics
CenterforConstructiveApproximation
VanderbiltUniversity

Nashville,TN,USA

ARTHURDAVIDSNIDER
DepartmentofElectricalEngineering
UniversityofSouthFlorida
Tampa,FL,USA
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LibraryofCongressCataloging-in-PublicationData:
Saff,E.B.,1944–
Fundamentalsofmatrixanalysiswithapplications/EdwardBarrySaff,CenterforConstructiveApproximation,

VanderbiltUniversity,Nashville,Tennessee,ArthurDavidSnider,DepartmentofElectricalEngineering,

UniversityofSouthFlorida,Tampa,Florida.
pages cm
Includesbibliographicalreferencesandindex.
ISBN978-1-118-95365-5(cloth)
1. Matrices. 2. Algebras,Linear. 3. Orthogonalizationmethods. 4. Eigenvalues.
I. Snider,ArthurDavid,1940– II. Title.
QA188.S1942015
(cid:2)
512.9434–dc23
2015016670

PrintedintheUnitedStatesofAmerica
10 9 8 7 6 5 4 3 2 1

1 2015
ToourbrothersHarveyJ.Saff,DonaldJ.Saff,andArthurHerndonSnider.Theyhave
set a high bar for us, inspired us to achieve, and extended helping hands when we

neededthemtoreachforthosegreaterheights.

EdwardBarrySaff

ArthurDavidSnider
CONTENTS

PREFACE ix

PARTI INTRODUCTION:THREEEXAMPLES 1
1 SystemsofLinearAlgebraicEquations 5
1.1 LinearAlgebraicEquations, 5

1.2 Matrix Representation of Linear Systems and the Gauss-Jordan
Algorithm, 17
1.3 TheCompleteGaussEliminationAlgorithm, 27

1.4 EchelonFormandRank, 38
1.5 ComputationalConsiderations, 46
1.6 Summary, 55

2 MatrixAlgebra 58

2.1 MatrixMultiplication, 58
2.2 SomePhysicalApplicationsofMatrixOperators, 69
2.3 TheInverseandtheTranspose, 76

2.4 Determinants, 86
2.5 ThreeImportantDeterminantRules, 100
2.6 Summary, 111
GroupProjectsforPartI
A. LUFactorization, 116
B. Two-PointBoundaryValueProblem, 118

C. ElectrostaticVoltage, 119
CONTENTS vii
D. Kirchhoff’sLaws, 120
E. GlobalPositioningSystems, 122
F. Fixed-PointMethods, 123

PARTII INTRODUCTION:THESTRUCTUREOFGENERAL

SOLUTIONSTOLINEARALGEBRAICEQUATIONS 129

3 VectorSpaces 133
3.1 GeneralSpaces,Subspaces,andSpans, 133
3.2 LinearDependence, 142
3.3 Bases,Dimension,andRank, 151

3.4 Summary, 164

4 Orthogonality 165
4.1 OrthogonalVectorsandtheGram–SchmidtAlgorithm, 165

4.2 OrthogonalMatrices, 174

4.3 LeastSquares, 180
4.4 FunctionSpaces, 190
4.5 Summary, 197
GroupProjectsforPartII

A. RotationsandReflections, 201
B. HouseholderReflectors, 201

C. InfiniteDimensionalMatrices, 202

PARTIII INTRODUCTION:REFLECTONTHIS 205
5 EigenvectorsandEigenvalues 209

5.1 EigenvectorBasics, 209

5.2 CalculatingEigenvaluesandEigenvectors, 217

5.3 SymmetricandHermitianMatrices, 225

5.4 Summary, 232
6 Similarity 233
6.1 SimilarityTransformationsandDiagonalizability, 233
6.2 PrincipleAxesandNormalModes, 244
6.3 SchurDecompositionandItsImplications, 257
6.4 TheSingularValueDecomposition, 264
6.5 ThePowerMethodandtheQRAlgorithm, 282

6.6 Summary, 290
viii CONTENTS
7 LinearSystemsofDifferentialEquations 293
7.1 First-OrderLinearSystems, 293
7.2 TheMatrixExponentialFunction, 306

7.3 TheJordanNormalForm, 316

7.4 MatrixExponentiationviaGeneralizedEigenvectors, 333

7.5 Summary, 339
GroupProjectsforPartIII
A. PositiveDefiniteMatrices, 342
B. HessenbergForm, 343

C. DiscreteFourierTransform, 344
D. ConstructionoftheSVD, 346
E. TotalLeastSquares, 348

F. FibonacciNumbers, 350

ANSWERSTOODDNUMBEREDEXERCISES 351
INDEX 393