Quaternion Matrix Computations

229 Pages·2018·10.153 MB·English

by  Musheng Wei, Ying Li, Fengxia Zhang, Jianli Zhao

MATHEMATICS RESEARCH DEVELOPMENTS

Q M
UATERNION ATRIX
C
OMPUTATIONS

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M R
ATHEMATICS ESEARCH
D
EVELOPMENTS

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MATHEMATICS RESEARCH DEVELOPMENTS

Q M
UATERNION ATRIX
C
OMPUTATIONS

MUSHENG WEI
YING LI
FENGXIA ZHANG
AND
JIANLI ZHAO
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Published by Nova Science Publishers, Inc. † New York
Contents
Preface ix
Acknowledgments xi
Notations xiii
1 Preliminaries 1
1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2. Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3. QuaternionMatrices . . . . . . . . . . . . . . . . . . . . . . . 4
1.4. EigenvalueProblem . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5. Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.5.1. VectorNorms . . . . . . . . . . . . . . . . . . . . . . 14
1.5.2. MatrixNorms . . . . . . . . . . . . . . . . . . . . . . 17
1.6. GeneralizedInverses . . . . . . . . . . . . . . . . . . . . . . . 23
1.7. Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.7.1. IdempotentMatricesandProjections. . . . . . . . . . . 26
1.7.2. OrthogonalProjections . . . . . . . . . . . . . . . . . . 29
1.7.3. GeometricMeaningsofAA† andA†A . . . . . . . . . . 30
1.8. PropertiesofRealRepresentationMatrices . . . . . . . . . . . 31
2 ComputingMatrixDecompositions 35
2.1. ElementaryMatrices . . . . . . . . . . . . . . . . . . . . . . . 36
2.2. TheQuaternionLUDecomposition . . . . . . . . . . . . . . . 45
2.3. TheQuaternionLDLH andCholesky
Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.4. TheQuaternionQRDecomposition . . . . . . . . . . . . . . . 58
vi Contents
2.4.1. TheQuaternionHouseholderQRD . . . . . . . . . . . 59
2.4.2. TheGivensQRD . . . . . . . . . . . . . . . . . . . . . 66
2.4.3. TheModifiedGram-SchimitScheme . . . . . . . . . . 67
2.4.4. CompleteOrthogonalDecomposition . . . . . . . . . . 69
2.5. TheQuaternionSVD . . . . . . . . . . . . . . . . . . . . . . . 69
3 LinearSystemandGeneralizedLeastSquaresProblems 79
3.1. LinearSystem . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.2. TheLinearLeastSquaresProblem . . . . . . . . . . . . . . . . 82
3.2.1. TheLSProblemandItsEquivalentProblems . . . . . . 82
3.2.2. TheRegularizationoftheLSProblem . . . . . . . . . . 85
3.2.3. SomeMatrixEquations . . . . . . . . . . . . . . . . . 87
3.3. TheTotalLeastSquaresProblem . . . . . . . . . . . . . . . . . 92
3.4. TheEqualityConstrainedLeastSquaresProblem . . . . . . . . 96
4 DirectMethodsforSolvingLinearSystemand
GeneralizedLSProblems 103
4.1. DirectMethodsforLinearSystem . . . . . . . . . . . . . . . . 104
4.2. DirectMethodsfortheLSProblem. . . . . . . . . . . . . . . . 105
4.3. DirectMethodsfortheTLSProblem . . . . . . . . . . . . . . . 107
4.4. DirectMethodsfortheLSEProblem . . . . . . . . . . . . . . . 108
4.5. SomeMatrixEquations . . . . . . . . . . . . . . . . . . . . . . 111
5 IterativeMethodsforSolvingLinearSystemand
GeneralizedLSProblems 119
5.1. BasicKnowledge . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.1.1. TheChebyshevPolynomials . . . . . . . . . . . . . . . 120
5.1.2. The Range of Eigenvaluesof Real Symmetric Tridiag-
onalMatrices . . . . . . . . . . . . . . . . . . . . . . . 121
5.2. IterativeMethodsforLinearSystem . . . . . . . . . . . . . . . 122
5.2.1. BasicTheoryofSplittingIterativeMethod . . . . . . . 122
5.2.2. TheKrylovSubspaceMethods . . . . . . . . . . . . . . 132
5.3. IterativeMethodsfortheLSProblem . . . . . . . . . . . . . . 137
5.3.1. SplittingIterativeMethods . . . . . . . . . . . . . . . . 138
5.3.2. TheKrylovSubspaceMethods . . . . . . . . . . . . . . 143
5.3.3. Preconditioning Hermitian-Skew Hermitian Splitting
IterationMethods . . . . . . . . . . . . . . . . . . . . . 149
Contents vii
5.4. IterativeMethodsfortheTLSProblem . . . . . . . . . . . . . . 151
5.4.1. ThePartialSVDMethod . . . . . . . . . . . . . . . . . 151
5.4.2. BidiagonalizationMethod . . . . . . . . . . . . . . . . 152
5.5. SomeMatrixEquations . . . . . . . . . . . . . . . . . . . . . . 154
6 ComputationsofQuaternionEigenvalueProblems 157
6.1. QuaternionHermitianRightEigenvalueProblem . . . . . . . . 158
6.1.1. ThePowerMethodandInversePowerMethodfor
QuaternionHermitianRightEigenvalueProblem . . . . 158
6.1.2. RealStructure-PreservingAlgorithmofHermitianQR
AlgorithmforHermitianRightEigenvalueProblem . . 164
6.1.3. Real Structure-Preserving Algorithm of the Jacobi
MethodforHermitianRightEigenvalueProblem . . . . 168
6.1.4. SubspaceMethods . . . . . . . . . . . . . . . . . . . . 175
6.2. QuaternionNon-HermitianRightEigenvalue
Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
6.2.1. ThePowerMethodandtheInversePowerMethod . . . 177
6.2.2. TheQuaternionQRAlgorithmforQuaternion
Non-HermitianRightEigenvalueProblem . . . . . . . . 185
References 189
AbouttheAuthors 207
Index 209

Preface
In 1843, Sir William Rowan Hamilton (1805-1865) introduced quaternion as
he tried to extend complex numbers to higher spatial dimensions, and then he
spenttherestofhislifeobsessedwiththemandtheirapplications.Nevertheless
he probably never thoughtthatone day in the future the quaternionhe had in-
ventedwouldbeusedinquaternionicquantummechanics(qQM),colorimage
processingandmanyotherfields.
About 100 years later, Finkelstein et al built the foundationsof qQM and
gaugetheories. Theirfundamentalworksledtoarenewedinterestinalgebriza-
tionandgeometrizationofphysicaltheoriesbynon-commutativefields. Among
the numerous references on this subject, the important paper of Horwitz and
Biedenharn showed that the assumption of a complex projection of the scalar
product, also called complex geometry Rembielinski,permits the definitionof
a suitabletensor product between single-particle quaternionicwave functions.
Now quaternionbecomes to playan importantrole in many applicationfields,
such as special relativity, group representations, non-relativistic and relativis-
tic dynamics, field theory, Lagrangian formalism, electro weak model, grand
unificationtheoriesandsoon.
In 1996, Sangwine proposed to encode three channel components of an
RGB image on the three imaginary parts of a pure quaternion. Thus, a color
image can be represented by a pure imaginary quaternion matrix. Since then,
quaternion representation of a color image has attracted great attention. With
therapiddevelopmentofapplicationsintheabovedisciplines,itisnecessaryto
furtherstudynumericalcomputationsofquaternionmatrices. SangwineandLe
BihanestablishedaplausibleQuaternionToolboxsoftwareforMatlab.
Duringtherecentyears,therearetwokindsofmethodsproposedforquater-
nionmatrixcomputationsbasedonrealarithmeticoperations. Thefirstkindof
methodistodirectlyperform computationsonthereal representationmatrices