Linear algbra for computational sciences and engineering

Free ebooks ==> www.Ebook777.com Ferrante Neri Linear Algebra for Computational Sciences and Engineering www.Ebook777.com Free ebooks ==> www.Ebook777.com Linear Algebra for Computational Sciences and Engineering www.Ebook777.com Ferrante Neri Linear Algebra for Computational Sciences and Engineering Foreword by Alberto Grasso 123 Ferrante Neri Centre for Computational Intelligence De Montfort University Leicester UK and University of Jyväskylä Jyväskylä Finland ISBN 978-3-319-40339-7 DOI 10.1007/978-3-319-40341-0 ISBN 978-3-319-40341-0 (eBook) Library of Congress Control Number: 2016941610 © Springer International Publishing Switzerland 2016 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part 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 or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use 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 authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland Free ebooks ==> www.Ebook777.com We can only see a short distance ahead, but we can see plenty there that needs to be done Alan Turing www.Ebook777.com Foreword Linear Algebra in Physics The history of linear algebra can be viewed within the context of two important traditions The first tradition (within the history of mathematics) consists of the progressive 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, permutations Complex numbers and vector analysis belong to this tradition Within the development of mathematics, the one was concerned not so much about solving specific equations, but mostly about addressing general and fundamental questions 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 operations thus allowing to optimize some non-linear mathematical problems As a first example, Lattices were generalizations of order relations on algebraic spaces, such as set inclusion in set theory and inequality in the familiar number systems (N, Z, Q, and R) As a second example, Boolean algebra generalized the operations of 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 This approach to logic as an algebraic structure was much similar as the Descartes’ algebra approach to the geometry Set theory and logic have been further advanced in the past centuries In particular, Hilbert attempted to build up mathematics by using symbolic logic in a way that could prove its consistency On the other hand, Gödel proved that in any mathematical system there will always be statements that 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 physical reality This tradition played a role in the Greek geometry’s bearing and its following application to physical problems When observing the space around us, vii viii Foreword we always suppose the existence of a reference frame, identified with an ideal “rigid body”, in the part of the universe in which the system we want to study evolves (e.g a three axes’ system having the Sun as their origin and direct versus three fixed 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 frames have to be intuitively considered distinct if the correspondent “rigid bodies” are in relative motion Therefore, it is important to fix the links (Linear Transformations) between the kinematic entities associated to the same motion but related to two different reference frames (Galileo’s Relativity) In the XVII and XVIII centuries, some physical entities needed a new representation This necessity made the above-mentioned two traditions converged by adding quantities as velocity, force, momentum and acceleration (vectors) to the traditional quantities as mass and time (scalars) Important ideas led to the vectors’ major systems: the forces’ parallelogram concept by Galileo, the situations geometry and calculus concepts by Leibniz and by Newton and the complex numbers’ geometrical representation Kinematics studies the motion of bodies in space and in time independently on the causes which provoke it In classical physics, the role of time is reduced to that of a parametric independent variable It needs also to choose a model for the body (or bodies) whose motion one wants to study The fundamental 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 quantities considered in a particular problem) The motion of a point is represented by a curve in the tridimensional Euclidean affine space A second fundamental model is the “rigid body” one, adopted for those extended bodies whose component particles 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 respectively represent orientation and rotation of a body in three dimensions Thus, it was already clear how a simple algebra should meet the needs of the physicists in order to efficiently describe objects in space and in time (in particular, their Dynamical Symmetries and the corresponding Conservation Laws) and the properties of space-time itself Furthermore, the principal characteristic of a simple algebra had to be its Linearity (or at most its multi-Linearity) During the latter part of the XIX century, Gibbs based his three dimensional vector algebra on some ideas 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 introduced Clifford algebra’s matrix representations for physical reasons: Pauli for describing the electron spin, while Dirac for accommodating both the electron spin and the special relativity Each algebraic system is widely used in Contemporary Physics and is a fundamental part of representing, interpreting, and understanding the nature 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 is the fundamental output (linear response) of each basic input Ik Then, both an arbitrary input as its own response can be written as a linear combination of the basic ones, i.e I ẳ c1 I1 ỵ ỵ ck Ik and O ẳ c1 O1 ỵ þ ck Ok Decoupling Principle If a system of coupled differential equations (or difference equations) involves a diagonalizable square matrix A, then it is useful to consider new variables x0k ẳ Uxk with k N; k nị, where U is an Unitary matrix and x0k is an orthogonal eigenvectors set (basis) of A Rewriting the equations in terms of the x0k , the one discovers that each eigenvectors evolution is 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 x0k as a function of time, it is also possible to get xk as a function of time (xk ¼ UÀ1 x0k ) 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, if one does not take into account some deep problems related to the possible presence of resonances) Symmetry Principle If A is a diagonal matrix representing a linear transformation of a physical system’s state and x0 k its eigenvectors set, each unitary transformation satisfying the matrix equation UAUÀ1 ¼ A (or UA ¼ AU) is called “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 symmetric 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 aspects while neglecting others important issues For example, Galileo founded the Mechanics neglecting friction, despite its important effect on mechanics The statement of Galileo’s Inertia Law (Newton’s First Law, i.e “An object not affected by forces moves with constant velocity”) is a pure abstraction and it is approximately 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 For example, thanks to Newton’s Second Law, much of classical physics is expressed in terms of second order ordinary differential equations’ systems If the force is a position’s linear function, the resulting equations are linear (m ddt2x ¼ ÀAx, 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 dominant for small oscillations A detailed treatment of coupled small oscillations is possible by obtaining a diagonal matrix of the coefficients in N coupled differential equations by finding the eigenvalues and the eigenvectors of the Lagrange’s equations for coupled oscillators In classical mechanics, another example of linearisation 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 electromagnetism have an infinite number of degrees of freedom (i.e the value of the field at each point) but the Superposition Principle and the Decoupling Principle still apply 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 inertia and electromagnetic field tensors), but also in engineering (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 In this case, the Three Neutrinos Masses Matrix is not diagonal and not normal in the so called Gauge States’ basis However, through a bi-unitary transformation (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 and many problems are reduced to obtain a diagonal Hamiltonian operator 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’ framework) researchers encounter symmetric, real positive definite and thus diagonalizable 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, the variance’s notion is so generalized to multiple dimension The geometrical symmetry’s notion played an essential part in constructing simplified theories governing the motion of galaxies and the microstructure of matter (quarks’ motion confined inside the hadrons and leptons’ motion) It was not until the Einstein’s era that the discovery of the space-time symmetries of the fundamental laws and the meaning of their relations to the 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 Free ebooks ==> www.Ebook777.com Foreword xi constitutes a state whose symmetry properties are to be studied The higher its “degree of symmetry” (and the number of conditions defining the state is reduced) 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 that an equation of order n ! cannot, in general, be solved by algebraic methods He did this by showing that the functional relations among the roots of an equation 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 transformation of a group in the theory of differential equations In the 1920s, Weyl and Wigner recognized that certain group theory’s methods could be used as a powerful analytical tool in Quantum Physics In particular, the essential role played by Lie’s groups, e.g rotation isomorphic groups SOð3Þ and SU ð2Þ, was first emphasized by Wigner Their ideas have been used in many contemporary physics’ 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 observed to be broken In physics, two different kinds of symmetry breakdown are 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 Symmetry Breaking” In this sense, the underlying laws of a system maintain their form (Lagrange’s Equations are invariant) under a symmetry transformation, but the 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 the related Levi-Civita symbolism, one gains an important and easy calculation tool 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 www.Ebook777.com ... ebooks ==> www.Ebook777.com Linear Algebra for Computational Sciences and Engineering www.Ebook777.com Ferrante Neri Linear Algebra for Computational Sciences and Engineering Foreword by Alberto Grasso... laws and regulations and therefore free for general use 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... 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 dominant for small oscillations