Determinants and their applications in mathematical physics PDF
Preview Determinants and their applications in mathematical physics
Applied Mathematical Sciences Volume 134 Editors J.E. Marsden L. Sirovich Advisors S. Antman J.K. Hale P. Holmes T. Kambe J. Keller K. Kirchgassner B.J. Matkowsky C.S. Peskin Springer NewYork Berlin Heidelberg Barcelona Budapest HongKong London Milan Paris Singapore Tokyo Applied Mathematical Sciences 1. fohn: PartialDifferentialEquations,4thed. 34. Kevorkian/Cole:PerturbationMethodsin 2. Sirovich:TechniquesofAsymptoticAnalysis. AppliedMathematics. 3. Hale: TheoryofFunctionalDifferential 35. Carr: ApplicationsofCentreManifoldTheory. Equations,2nded. 36. Bengtsson/GhiIlKiillen:DynamicMeteorology: 4. Percus: CombinatorialMethods. DataAssimilationMethods. 5. vonMises/Friedrichs: FluidDynamics. 37. Saperstone: SemidynamicalSystemsinInfinite 6. Freiberger/Grenander: AShortCoursein DimensionalSpaces. ComputationalProbabilityandStatistics. 38. Lichtenberg/Lieberman:RegularandChaotic 7. Pipkin: LecturesonViscoelasticityTheory. Dynamics,2nded. 8. Giacoglia: PerturbationMethodsinNon-linear 39. Piccini/StampacchiaIVidossich:Ordinary Systems. DifferentialEquationsinR". 9. Friedrichs:SpectralTheoryofOperatorsin 40. Naylor/Sell:LinearOperatorTheoryin HilbertSpace. EngineeringandScience. 10. Stroud:NumericalQuadratureandSolutionof 41. Sparrow:TheLorenzEquations: Bifurcations, OrdinaryDifferentialEquations. Chaos,andStrangeAttractors. II. Wolovich: LinearMultivariableSystems. 42. Guckenheimer/Holmes: NonlinearOscillations, 12. Berkovitz:OptimalControlTheory. DynamicalSystems,andBifurcationsofVector 13. Bluman/Cole:SimilarityMethodsfor Fields. DifferentialEquations. 43. OckendonITaylor: lnviscidFluidFlows. 14. Yoshizawa: StabilityTheoryandtheExistence 44. Pazy: SemigroupsofLinearOperatorsand ofPeriodicSolutionandAlmostPeriodic ApplicationstoPartialDifferentialEquations. Solutions. 45. Glashojf/Gustafson: LinearOperationsand 15. Braun: DifferentialEquationsandTheir Approximation: AnIntroductiontothe Applications,3rded. TheoreticalAnalysisandNumericalTreatment 16. Lef,chetz: ApplicationsofAlgebraicTopology. ofSemi-InfinitePrograms. 17. CollatzIWetterling:OptimizationProblems. 46. Wilcox: ScatteringTheoryforDiffraction 18 Grenander: PatternSynthesis:Lecturesin Gratings. PatternTheory,Vol.1. 47. Haleetale AnIntroductiontoInfinite 19. Marsden/McCracken: HopfBifurcationandIts DimensionalDynamicalSystems-Geometric Applications. Theory. 20. Driver:OrdinaryandDelayDifferential 48. Murray: AsymptoticAnalysis. Equations. 49. Ladyzhenskaya:TheBoundary-ValueProblems 21. Courant/Friedrichs: SupersonicFlowandShock ofMathematicalPhysics. Waves. 50. Wilcox: SoundPropagationinStratifiedFluids. 22. Rouche/Habets/Laloy:StabilityTheoryby 51. Golubitsky/Schaejfer: BifurcationandGroupsin Liapunov'sDirectMethod. BifurcationTheory,Vol.1. 23. Lamperti:StochasticProcesses: ASurveyofthe 52. Chipot: Variational InequalitiesandFlowin MathematicalTheory. PorousMedia. 24. Grenander: PatternAnalysis: LecturesinPattern 53. Majda: CompressibleFluidFlowandSystemof Theory, Vol. ll. ConservationLawsinSeveralSpaceVariables. 25. Davies: IntegralTransformsandTheir 54. Wasow: LinearTurningPointTheory. Applications, 2nded. 55. Yosida: OperationalCalculus:ATheoryof 26. Kushner/Clark: StochasticApproximation Hyperfunctions. MethodsforConstrainedandUnconstrained 56. Chang/Howes:NonlinearSingularPerturbation Systems. Phenomena:TheoryandApplications. 27. deBoor:APracticalGuidetoSplines. 57. Reinhardt: AnalysisofApproximationMethods 28. Keilson:MarkovChainModels-Rarityand forDifferentialandIntegralEquations. Exponentiality. 58. Dwoyer/HussainWoigt(eds): Theoretical 29. de Veubeke: ACourseinElasticity. ApproachestoTurbulence. 30. Shiarycki: GeometricQuantizationandQuantum 59. Sanders/Verhulst:AveragingMethodsin Mechanics. NonlinearDynamicalSystems. 31. Reid:SturmianTheoryforOrdinaryDifferential 60. GhillChildress:TopicsinGeophysical Equations. Dynamics: AtmosphericDynamics.Dynamo 32. Meis/Markowitz: NumericalSolutionofPartial TheoryandClimateDynamics. DifferentialEquations. 33 Grenander: RegularStructures: Lecturesin PatternTheory, Vol.m. (continuedfollowing index) Robert Vein Paul Dale Determinants and Their Applications in Mathematical Physics , Springer RobertVein PaulDale Aston University DepartmentofComputerScience and Applied Mathematics Aston Triangle Birmingham B47ET UK Editors J.E. Marsden L. Sirovich Control and Dynamical Systems, 107-81 DivisionofApplied Mathematics CaliforniaInstitute ofTechnology Brown University Pasadena, CA 91125 Providence, RI 02912 USA USA MathematicsSubjectClassification (1991): 15A15,33A65,34A34, 83C99, 35Q20 LibraryofCongressCataloging-in-PublicationData Vein, Robert. Determinants andtheirapplicationsin mathematicalphysics/ byRobertVeinandPaulDale p. cm.-(Appliedmathematicalsciences; 134) Includes bibliographicalreferences and index. ISBN0-387-98558-1 (alk. paper) 1.Determinants. 2.Mathematical physics. I.Dale,Paul. II.Title. III.Series:Applied mathematicalsciences (Springer-Verlag New YorkInc.); v. 134. QAI.A647 [QC20.7.D45] 510s-dc21 [530.15] 98-7731 © 1999Springer-VerlagNew York, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the writtenpermissionofthepublisher(Springer-VerlagNewYork,Inc., 175FifthAvenue,NewYork, NY 10010, USA), exceptfor briefexcerpts in connection with reviews orscholarly analysis. Use inconnection with any form ofinformation storage and retrieval, electronic adaptation, computer software,orbysimilarordissimilarmethodology nowknown orhereafterdevelopedisforbidden. Theuseofgeneraldescriptivenames, trade names,trademarks,etc.,in thispublication,evenifthe former arenotespeciallyidentified, isnottobetaken as a signthatsuch names, as understoodby theTradeMarksandMerchandiseMarks Act,may accordinglybeusedfreely by anyone. ISBN0-387-98558-1 Springer-Verlag NewYork Berlin Heidelberg SPIN 10681036 Preface The last treatise on the theory of determinants, by T. Muir, revised and enlarged by W.H. Metzler, was published by Dover Publications Inc. in 1960. It is an unabridged and corrected republication of the edition origi- nallypublishedbyLongman,GreenandCo.in1933andcontainsapreface by Metzler dated 1928. The Table of Contents of this treatise is given in Appendix 13. A small number of other books devoted entirely to determinants have beenpublishedinEnglish,buttheycontainlittleifanythingofimportance that was not known to Muir and Metzler. A few have appeared in German and Japanese. In contrast, the shelves of every mathematics library groan under the weight of books on linear algebra, some of which contain short chapters on determinants but usually only on those aspects of the subject whichareapplicabletothechaptersonmatrices.Thereappearstobetacit agreement among authorities on linear algebra that determinant theory is important only as a branch of matrix theory. In sections devoted entirely to the establishment of a determinantal relation, many authors define a determinant by first defining a matrix M and then adding the words: “Let detM be the determinant of the matrix M” as though determinants have no separate existence. This belief has no basis in history. The origins of determinants can be traced back to Leibniz (1646–1716) and their prop- erties were developed by Vandermonde (1735–1796), Laplace (1749–1827), Cauchy(1789–1857)andJacobi(1804–1851)whereasmatriceswerenotin- troduced until the year of Cauchy’s death, by Cayley (1821–1895). In this book, most determinants are defined directly. vi Preface It may well be perfectly legitimate to regard determinant theory as a branch of matrix theory, but it is such a large branch and has such large and independent roots, like a branch of a banyan tree, that it is capable of leading an independent life. Chemistry is a branch of physics, but it is sufficiently extensive and profound to deserve its traditional role as an independent subject. Similarly, the theory of determinants is sufficiently extensive and profound to justify independent study and an independent book. This book contains a number of features which cannot be found in any otherbook.Prominentamongthesearetheextensiveapplicationsofscaled cofactors and column vectors and the inclusion of a large number of rela- tions containing derivatives. Older books give their readers the impression that the theory of determinants is almost entirely algebraic in nature. If the elements in an arbitrary determinant A are functions of a continuous variablex,thenApossessesaderivativewithrespecttox.Theformulafor this derivative has been known for generations, but its application to the solution of nonlinear differential equations is a recent development. Thefirstfivechaptersarepurelymathematicalinnatureandcontainold andnewproofsofseveraloldtheoremstogetherwithanumberoftheorems, identities, and conjectures which have not hitherto been published. Some theorems, both old and new, have been given two independent proofs on the assumption that the reader will find the methods as interesting and important as the results. Chapter 6 is devoted to the applications of determinants in mathemat- ical physics and is a unique feature in a book for the simple reason that these applications were almost unknown before 1970, only slowly became known during the following few years, and did not become widely known until about 1980. They naturally first appeared in journals on mathemat- ical physics of which the most outstanding from the determinantal point of view is the Journal of the Physical Society of Japan. A rapid scan of Section 15A15 in the Index of Mathematical Reviews will reveal that most pure mathematicians appear to be unaware of or uninterested in the out- standingcontributionstothetheoryandapplicationofdeterminantsmade inthecourseofresearchintoproblemsinmathematicalphysics.Theseusu- allyappearinSection35QoftheIndex.Puremathematiciansarestrongly recommended to make themselves acquainted with these applications, for they will undoubtedly gain inspiration from them. They will find plenty of scope for purely analytical research and may well be able to refine the techniques employed by mathematical physicists, prove a number of con- jectures, and advance the subject still further. Further comments on these applications can be found in the introduction to Chapter 6. Thereappearstobenogeneralagreementonnotationamongwriterson determinants. We use the notion A =|a | and B =|b | , where i and n ij n n ij n j are row and column parameters, respectively. The suffix n denotes the order of the determinant and is usually reserved for that purpose. Rejecter Preface vii minors of A are denoted by M(n), etc., retainer minors are denoted by n ij N , etc., simple cofactors are denoted by A(n), etc., and scaled cofactors ij ij are denoted by Aij, etc. The n may be omitted from any passage if all the n determinants which appear in it have the same order. The letter D, some- times with a suffix x, t, etc., is reserved for use as a differential operator. The letters h, i, j, k, m, p, q, r, and s are usually used as integer param- eters. The letter l is not used in order to avoid confusion with the unit integer. Complex numbers appear in some sections and pose the problem of conflicting priorities. The notation ω2 =−1 has been adopted since the letters i and j are indispensable as row and column parameters, respec- tively, in passages where a large number of such parameters are required. Matrices are seldom required, but where they are indispensable, they ap- pear in boldface symbols such as A and B with the simple convention A=detA, B =detB, etc. The boldface symbols R and C, with suffixes, arereservedforuseasrowandcolumnvectors,respectively.Determinants, their elements, their rejecter and retainer minors, their simple and scaled cofactors,theirrowandcolumnvectors,andtheirderivativeshaveallbeen expressed in a notation which we believe is simple and clear and we wish to see this notation adopted universally. TheAppendixconsistsmainlyofnondeterminantalrelationswhichhave been removed from the main text to allow the analysis to proceed without interruption. The Bibliography contains references not only to all the authors men- tioned in the text but also to many other contributors to the theory of determinants and related subjects. The authors have been arranged in al- phabetical order and reference to Mathematical Reviews, Zentralblatt fu¨r Mathematik,andPhysicsAbstractshavebeenincludedtoenablethereader whohasnoeasyaccesstojournalsandbookstoobtainmoredetailsoftheir contents than is suggested by their brief titles. The true title of this book is The Analytic Theory of Determinants with Applications to the Solutions of Certain Nonlinear Equations of Mathe- matical Physics, which satisfies the requirements of accuracy but lacks the virtueofbrevity.Chapter1beginswithabriefnoteonGrassmannalgebra and then proceeds to define a determinant by means of a Grassmann iden- tity.Later,theLaplaceexpansionandafewotherrelationsareestablished by Grassmann methods. However, for those readers who find this form of algebra too abstract for their tastes or training, classical proofs are also given. Most of the contents of this book can be described as complicated applications of classical algebra and differentiation. In a book containing so many symbols, misprints are inevitable, but we hope they are obvious and will not obstruct our readers’ progress for long. All reports of errors will be warmly appreciated. We are indebted to our colleague, Dr. Barry Martin, for general advice oncomputersandforinvaluableassistanceinalgebraiccomputingwiththe viii Preface Maple system on a Macintosh computer, especially in the expansion and factorization of determinants. We are also indebted by Lynn Burton for the most excellent construction and typing of a complicated manuscript in Microsoft Word programming language Formula on a Macintosh computer in camera-ready form. Birmingham, U.K. P.R. Vein P. Dale Contents Preface v 1 Determinants, First Minors, and Cofactors 1 1.1 Grassmann Exterior Algebra . . . . . . . . . . . . . . . . . 1 1.2 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.3 First Minors and Cofactors . . . . . . . . . . . . . . . . . . 3 1.4 The Product of Two Determinants — 1. . . . . . . . . . . 5 2 A Summary of Basic Determinant Theory 7 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Row and Column Vectors . . . . . . . . . . . . . . . . . . . 7 2.3 Elementary Formulas . . . . . . . . . . . . . . . . . . . . . 8 2.3.1 Basic Properties . . . . . . . . . . . . . . . . . . . . 8 2.3.2 Matrix-Type Products Related to Row and Column Operations . . . . . . . . . . . . . . . . . . 10 2.3.3 First Minors and Cofactors; Row and Column Expansions . . . . . . . . . . . . . . . . . . . . . . . 12 2.3.4 Alien Cofactors; The Sum Formula . . . . . . . . . 12 2.3.5 Cramer’s Formula . . . . . . . . . . . . . . . . . . . 13 2.3.6 The Cofactors of a Zero Determinant . . . . . . . . 15 2.3.7 The Derivative of a Determinant . . . . . . . . . . 15 3 Intermediate Determinant Theory 16 3.1 Cyclic Dislocations and Generalizations . . . . . . . . . . . 16 3.2 Second and Higher Minors and Cofactors . . . . . . . . . . 18
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