Linear Algebra for Computational Sciences and Engineering PDF
Preview Linear Algebra for Computational Sciences and Engineering
Ferrante Neri Linear Algebra for Computational Sciences and Engineering Linear Algebra for Computational Sciences and Engineering Ferrante Neri Linear Algebra for Computational Sciences and Engineering Foreword by Alberto Grasso 123 Ferrante Neri Centrefor Computational Intelligence De MontfortUniversity Leicester UK and University of Jyväskylä Jyväskylä Finland ISBN978-3-319-40339-7 ISBN978-3-319-40341-0 (eBook) DOI 10.1007/978-3-319-40341-0 LibraryofCongressControlNumber:2016941610 ©SpringerInternationalPublishingSwitzerland2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor foranyerrorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAGSwitzerland We can only see a short distance ahead, but we can see plenty there that needs to be done Alan Turing Foreword Linear Algebra in Physics The history of linear algebra can be viewed within the context of two important traditions. Thefirsttradition(withinthehistoryofmathematics)consistsoftheprogressive broadening of the concept of number so to include not only positive integers, but also negative numbers, fractions, algebraic and transcendental irrationals. Moreover, the symbols in the equations became matrices, polynomials, sets, per- mutations. Complex numbers and vector analysis belong to this tradition. Within thedevelopmentofmathematics,theonewasconcernednotsomuchaboutsolving specificequations,butmostlyaboutaddressinggeneralandfundamentalquestions. The latter were approached by extending the operations and the properties of sum and multiplication from integers to other linear algebraic structures. Different algebraic structures (Lattices and Boolean algebra) generalized other kinds of operationsthusallowingtooptimizesomenon-linearmathematicalproblems.Asa first example, Lattices were generalizations of order relations on algebraic spaces, suchassetinclusioninsettheoryandinequalityinthefamiliarnumbersystems(N, Z,Q,andR).Asasecondexample,Booleanalgebrageneralizedtheoperationsof intersection and union and the Principle of Duality (De Morgan’s Relations), already valid in set theory, to formalize the logic and the propositions’ calculus. ThisapproachtologicasanalgebraicstructurewasmuchsimilarastheDescartes’ algebraapproachtothegeometry.Settheoryandlogichavebeenfurtheradvanced in the past centuries. In particular, Hilbert attempted to build up mathematics by using symbolic logicin a way that could prove its consistency. On theother hand, Gödelprovedthatinanymathematicalsystemtherewillalwaysbestatementsthat can never be proven either true or false. The second tradition (within the history of physical science) consists of the search for mathematical entities and operations that represent aspects of the physicalreality.ThistraditionplayedaroleintheGreekgeometry’sbearingandits following application to physical problems. When observing the space around us, vii viii Foreword wealwayssupposetheexistenceofareferenceframe,identifiedwithanideal“rigid body”, in the part of the universe in which the system we want to study evolves (e.g.athreeaxes’systemhavingtheSunastheiroriginanddirectversusthreefixed stars). This is modelled in the so called “Euclidean affine space”. A reference frame’s choice is purely descriptive, at a purely kinematic level. Two reference frameshavetobeintuitivelyconsidereddistinctifthecorrespondent“rigidbodies” are in relative motion. Therefore, it is important to fix the links (Linear Transformations)betweenthekinematicentitiesassociatedtothesamemotionbut related to two different reference frames (Galileo’s Relativity). In the XVII and XVIII centuries, some physical entities needed a new repre- sentation. This necessity made the above-mentioned two traditions converged by adding quantities as velocity, force, momentum and acceleration (vectors) to the traditionalquantitiesasmassandtime(scalars).Importantideasledtothevectors’ major systems: the forces’ parallelogram concept by Galileo, the situations geom- etry and calculus concepts by Leibniz and by Newton and the complex numbers’ geometricalrepresentation.Kinematicsstudiesthemotionofbodiesinspaceandin timeindependentlyonthecauseswhichprovokeit.Inclassicalphysics,theroleof timeisreducedtothatofaparametricindependentvariable.Itneedsalsotochoose a model for the body (or bodies) whose motion one wants to study. The funda- mental and simpler model is that of point (useful only if the body’s extension is smaller than the extension of its motion and of the other important physical quantitiesconsideredinaparticularproblem).Themotionofapointisrepresented by a curve in the tridimensional Euclidean affine space. A second fundamental modelisthe“rigidbody”one,adoptedforthoseextendedbodieswhosecomponent particles do not change mutual distances during the motion. Later developments in Electricity, Magnetism, and Optics further promoted the use of vectors in mathematical physics. The XIX century marked the development of vector space methods, whose prototypes were the three-dimensional geometric extensive algebra by Grassmann and the algebra of quaternions by Hamilton to respectivelyrepresentorientationandrotationofabodyinthreedimensions.Thus, itwasalreadyclearhowasimplealgebrashouldmeettheneedsofthephysicistsin order to efficiently describe objects in space and in time (in particular, their Dynamical Symmetries and the corresponding Conservation Laws) and the prop- erties of space-time itself. Furthermore, the principal characteristic of a simple algebrahadtobeitsLinearity(oratmostitsmulti-Linearity).Duringthelatterpart oftheXIXcentury,Gibbsbasedhisthreedimensionalvectoralgebraonsomeideas by Grassmann and by Hamilton, while Clifford united these systems into a single geometric algebra (direct product of quaternions’ algebras). After, the Einsteins description of the four-dimensional continuum space-time (Special and General Relativity Theories) required a Tensor Algebra. In 1930s, Pauli and Dirac intro- duced Clifford algebra’s matrix representations for physical reasons: Pauli for describingtheelectronspin,whileDiracforaccommodatingboththeelectronspin and the special relativity. Each algebraic system is widely used in Contemporary Physics and is a fun- damentalpart ofrepresenting,interpreting, andunderstandingthenature.Linearity Foreword ix in physics is principally supported by three ideas: Superposition Principle, Decoupling Principle, and Symmetry Principle. Superposition Principle. Let us suppose to have a linear problem where each Ok isthefundamentaloutput(linearresponse)ofeachbasicinputIk.Then,bothan arbitrary input as its own response can be written as a linear combination of the basic ones, i.e. I ¼c1I1þ...þckIk and O¼c1O1þ...þckOk. Decoupling Principle. If a system of coupled differential equations (or differ- ence equations) involves a diagonalizable square matrix A, then it is useful to consider new variables x0k ¼Uxk with ðk 2N;1(cid:2)k(cid:2)nÞ, where U is an Unitary matrix and x0 is an orthogonal eigenvectors set (basis) of A. Rewriting the equa- k tions in terms of the x0, the one discovers that each eigenvectors evolution is k independent on the others and that the form of each equation depends only on the corresponding eigenvalue of A. By solving the equations so to get each x0 as a k function of time, it is also possible to get xk as a function of time (xk ¼U(cid:3)1x0k) . When A is not diagonalizable (not normal), the resulting equations for x are not completely decoupled (Jordan canonical form), but are still relatively easy (of course,ifonedoesnottakeintoaccountsomedeepproblemsrelatedtothepossible presence of resonances). Symmetry Principle. If A is a diagonal matrix representing a linear transfor- mation of a physical system’s state and x0k its eigenvectors set, each unitary transformationsatisfyingthematrixequationUAU(cid:3)1 ¼A(orUA¼AU)iscalled “Symmetry Transformation” for the considered physical system. Its deep meaning is to eventually change each eigenvector without changing the whole set of eigenvectors and their corresponding eigenvalues. Thus, special importance in computational physics is assumed by the standard methods for solving systems of linear equations: the procedures suited for sym- metric real matrices and the iterative methods converging fast when applied to matrix having its non-zero elements concentrated near the main diagonal (Diagonally Dominated Matrix). Physics has a very strong tradition about tending to focus on some essential aspectswhileneglectingothersimportantissues.Forexample,Galileofoundedthe Mechanics neglecting friction, despite its important effect on mechanics. The statementofGalileo’sInertiaLaw(Newton’sFirstLaw,i.e.“Anobjectnotaffected by forces moves with constant velocity”) is a pure abstraction and it is approxi- mately valid. While modelling, a popular simplification has been for centuries the search of a linear equation approximating the nature. Both Ordinary and Partial Linear Differential Equations appear through classical and quantum physics and even where the equations are non-linear, Linear Approximations are extremely powerful.Forexample,thankstoNewton’sSecondLaw,muchofclassicalphysics isexpressedintermsofsecondorderordinarydifferentialequations’systems.Ifthe forceisaposition’slinearfunction,theresultingequationsarelinear(md2x¼(cid:3)Ax, dt2 where A matrix not depending on x). Every solution may be written as a linear combination of the special solutions (oscillation’s normal modes) coming from eigenvectors of the A matrix. For nonlinear problems near equilibrium, the force x Foreword can always be expanded as a Taylor’s series and the leading (linear) term is dominantforsmalloscillations.Adetailedtreatmentofcoupledsmalloscillationsis possiblebyobtainingadiagonalmatrixofthecoefficientsinN coupleddifferential equations by finding the eigenvalues and the eigenvectors of the Lagrange’s equations for coupled oscillators. In classical mechanics, another example of lin- earisation consists of looking for the principal moments and principal axes of a solid body through solving the eigenvalues’ problem of a real symmetric matrix (Inertia Tensor). In the theory of continua (e.g. hydrodynamics, diffusion and thermal conduction, acoustic, electromagnetism), it is (sometimes) possible to convert a partial differential equation into a system of linear equations by employing the finite difference formalism. That ends up with a Diagonally Dominated coefficients’ Matrix. In particular, Maxwell’s equations of electromag- netism have an infinite number of degrees offreedom (i.e. the value of the field at eachpoint)buttheSuperpositionPrincipleandtheDecouplingPrinciplestillapply. The response to an arbitrary input is obtained as the convolution of a continuous basis of Dirac δ functions and the relevant Green’s function. Even without the differential geometry’s more advanced applications, the basic concepts of multilinear mapping and tensor are used not only in classical physics (e.g.inertiaandelectromagneticfieldtensors),butalsoinengineering(e.g.dyadic). In particle physics, it was important to analyse the problem of Neutrino Oscillations, formally related both to the Decoupling and the Superposition Principles.Inthiscase,theThreeNeutrinosMassesMatrixisnotdiagonalandnot normal in the so called Gauge States’ basis. However, through a bi-unitary trans- formation (one unitary transformation for each “parity” of the gauge states), it is possible to get the eigenvalues and their own eigenvectors (Mass States) which allow to render it diagonal. After this transformation, it is possible to obtain the Gauge States as a superposition (linear combination) of Mass States. Schrödinger’s Linear Equation governs the non relativistic quantum mechanics andmanyproblemsarereducedtoobtainadiagonalHamiltonianoperator.Besides, when studying the quantum angular momentum’s addition one considers Clebsch-Gordon coefficients related to an unitary matrix that changes a basis in a finite-dimensional space. In experimental physics and statistical mechanics (Stochastic methods’ frame- work) researchers encounter symmetric, real positive definite and thus diagonaliz- able matrices (so-called covariance or dispersion matrix). The elements of a covariance matrix in the i, j positions are the covariances between ith and jth elements of a random vector (i.e. a vector of random variables, each with finite variance).Intuitively,thevariance’snotionissogeneralizedtomultipledimension. The geometrical symmetry’s notion played an essential part in constructing simplified theories governing the motion of galaxies and the microstructure of matter(quarks’motionconfinedinsidethehadronsandleptons’motion).Itwasnot until the Einstein’s era that the discovery of the space-time symmetries of the fundamental laws and the meaning of their relations tothe conservation laws were fully appreciated, for example Lorentz Transformations, Noether’s Theorem and Weyl’s Covariance. An object with a definite shape, size, location and orientation Foreword xi constitutes a state whose symmetry properties are to be studied. The higher its “degree ofsymmetry” (and thenumberof conditions defining thestate isreduced) the greater is the number of transformations that leave the state of the object unchanged. While developing some ideas by Lagrange, by Ruffini and by Abel (among others), Galois introduced important concepts in group theory. This study showed thatanequationofordern(cid:4)5cannot,ingeneral,besolvedbyalgebraic methods. Hedidthisbyshowingthatthefunctionalrelationsamongtherootsofanequation have symmetries under the permutations of roots. In 1850s, Cayley showed that every finite group is isomorphic to a certain permutation group (e.g. the crystals’ geometrical symmetries are described in finite groups’ terms). Fifty years after Galois, Lie unified many disconnected methods of solving differential equations (evolved over about two centuries) by introducing the concept of continuous transformationofagroupinthetheoryofdifferentialequations.Inthe1920s,Weyl and Wigner recognized that certain group theory’s methods could be used as a powerfulanalyticaltoolinQuantumPhysics.Inparticular,theessentialroleplayed by Lie’s groups, e.g. rotation isomorphic groups SOð3Þ and SUð2Þ, was first emphasizedbyWigner.Theirideashavebeenusedinmanycontemporaryphysics’ branches which range from the Theory of Solids to Nuclear Physics and Particle Physics. In Classical Dynamics, the invariance of the equations of motion of a particle under the Galilean transformation is fundamental in Galileo’s relativity. The search for a linear transformation leaving “formally invariant” the Maxwell’s equations of Electromagnetism led to the discovery of a group of rotations in space-time (Lorentz transformation). Frequently, it is important to understand why a symmetry of a system is observedtobebroken. Inphysics,twodifferent kindsofsymmetrybreakdownare considered. If two states of an object are different (e.g. by an angle or a simple phase rotation) but they have the same energy, one refers to “Spontaneous SymmetryBreaking”.Inthissense, theunderlying laws ofasystem maintaintheir form(Lagrange’sEquationsareinvariant)underasymmetrytransformation,butthe system as a whole changes under such transformation by distinguishing between two or more fundamental states. This kind of symmetry breaking (for example) characterizes the ferromagnetic and the superconductive phases, where the Lagrange function (or the Hamiltonian function, representing the energy of the system) is invariant under rotations (in the ferromagnetic phase) and under a complex scalar transformation (in the superconductive phase). On the contrary, if the Lagrange function is not invariant under particular transformations, the so-called “Explicit Symmetry Breaking” occurs. For example, this happens when an external magnetic field is applied to a paramagnet (Zeeman’s Effect). Finally, by developing the determinants through the permutations’ theory and therelatedLevi-Civitasymbolism,onegainsanimportantandeasycalculationtool for modern differential geometry, with applications in engineering as well as in modern physics. This is the case in general relativity, quantum gravity, and string theory.
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