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106 II. Concepts

2 which, for a sufficiently fine partition, produces the integral enclosure ∈ 66 I 0.347400172649. Thus, it turns out that the result from simpson was completely wrong! This is one example of the impor- tance of rigorous computations. 0 3 Recent Developments

There is currently an ongoing effort within the IEEE community to standardize the implementation of inter- val arithmetic. The hope is that we will enable com- puter manufacturers to incorporate these types of com- –2 putations at the hardware level. This would remove –5 0 5 the large computational penalty incurred by repeat- Figure 1 Successively tighter enclosures of a graph. edly having to switch rounding modes—a task that central processing units were not designed to perform 2 Interval Analysis efficiently.

The inclusion principle (2) enables us to capture contin- Further Reading uous properties of a function, using only a finite num- Alefeld, G., and J. Herzberger. 1983. Introduction to Interval ber of operations. Its most important use is to explic- Computations. New York: Academic Press. itly bound discretization errors that naturally arise in Kulisch, U. W., and W. L. Miranker. 1981. Computer Arith- numerical algorithms. metic in Theory and Practice. New York: Academic Press. 3 As an example, consider the function f(x)= cos x+ Lerch, M., G. Tischler, J. Wolff von Gudenberg, W. Hof- sin x on the domain x = [−5, 5]. For any decompo- schuster, and W. Krämer. 2006. filib++, a fast inter- sition of the domain x into a finite set of subinter- val library supporting containment computations. ACM = n Transactions on Mathematical Software 32(2):299–324. vals x i=1 xi, we can form the set-valued graph Moore, R. E., R. B. Kearfott, and M. J. Cloud. 2009. Introduc- consisting of the pairs (x1,F(x1)),...,(xn,F(xn)).As tion to Interval Analysis. Philadelphia, PA: SIAM. the partition is made finer (that is, as maxi diam(xi) is Rump, S. M. 1999. INTLAB—INTerval LABoratory. In Devel- made smaller), the set-valued graph tends to the graph opments in Reliable Computing, edited by T. Csendes, of f (see figure 1). And, most importantly, every such pp. 77–104. Dordrecht: Kluwer Academic. set-valued graph contains the graph of f . Tucker, W. 2011. Validated Numerics: A Short Introduc- This way of incorporating the discretization errors tion to Rigorous Computations. Princeton, NJ: Princeton is extremely useful for quadrature, optimization, and University Press. equation solving. As one example, suppose we wish to = 8 + x compute the definite integral I 0 sin(x e ) dx. A MATLAB function simpson that implements a sim- II.21 Invariants and Conservation ple textbook adaptive Simpson quadrature algorithm Laws produces the following result. Mark R. Dennis

% Compute integral I with tolerance 1e-6. As important as the study of change in the mathemati- >> I = simpson(@(x) sin(x + exp(x)), 0, 8) cal representation of physical phenomena is the study I= of invariants. Physical laws often depend only on the 0.251102722027180 relative positions and times between phenomena, so certain physical quantities do not change; i.e., they are A(very naive) set-valued approach to quadrature is to invariant, under continuous translation or rotation of enclose the integral I via the spatial axes. Furthermore, as the spatial configura- n tion of a system evolves with time, quantities such as I ∈ F(x ) diam(x ), i i total energy may remain unchanged; that is, they are i=1

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II.21. Invariants and Conservation Laws 107 conserved. The study of invariants has been a remark- that physical phenomena do not depend on the overall ably successful approach to the mathematical formu- phase (argument) of this vector. Extension of Noether’s lation of physical laws, and the study of continuous theorem here leads to the conservation of electric symmetries and conservation laws—which are related charge, and extension to Yang–Mills theories provides by the result known as Noether’s theorem—has become other conserved quantities associated with the nuclear a systematic part of our description of physics over the forces studied in contemporary fundamental particle last century, from the atomic scale to the cosmic scale. physics. Other phenomena, such as the Higgs mecha- As an example of so-called Galilean invariance, New- nism (leading to the Higgs boson recently discovered ton’s force law keeps the same form when the velocity in high-energy experiments), are a consequence of the of the frame of reference (i.e., the coordinate system breaking of certain quantum symmetries in certain low- specified by x-, y-, and z-axes) is changed by adding energy regimes. Symmetry and symmetry breaking in a constant; this is equivalent to adding the same con- quantum theory are discussed briefly at the end of this stant velocity to all the particles in a mechanical sys- article. tem. Other quantities do change under such a veloc- Spatial vectors, such as r = (x,y,z), represent the 1 | |2 spatial distance between a chosen point and the ori- ity transformation, such as the kinetic energy 2 m v (for a particle of mass m and velocity v); however, for gin, and of course the vector between two such points an evolving, nondissipative system such as a bouncing, r2 − r1 is independent of translations of this origin. perfectly elastic rubber ball, the total energy is constant Similarly, the scalar product r2 ·r1 is unchanged under in time—that is, energy is conserved. rotation of the coordinate system by an orthogonal The development of our understanding of fundamen- matrix R, under which r → Rr. tal laws of dynamics can be interpreted by progres- Continuous groups of transformations such as trans- sively more sophisticated and general representations lation and rotation, and their matrix representations, of space and time themselves: ancient Greek physi- are an important tool used in calculations of invari- cal science assumed absolute space with a privileged ants. For example, the set of two-dimensional matrices cos θ − sin θ spatial point (the center of the Earth), through static sin θ cos θ , representing rotations through angles θ, Euclidean space where all spatial points are equiv- may be considered as a continuous one-parameter Abe- matrix expo- alent, through classical mechanics [IV.19] where lian group of matrices generated by the nential θA 0 −1 all inertial frames, moving at uniform velocity with [II.14] e , where A is the generator 10 . respect to each other, are equivalent according to The generator itself is found as the derivative of the Newton’s first law, to the modern theories of spe- original matrix with respect to θ, evaluated at θ = 0. cial and general relativity. The theory of relativity Translations are less obviously represented by matri- (both general and special) is motivated by Einstein’s ces; one approach is to append an extra dimension to principle of covariance, which is described below. the position vector with unit entry, such as (1,x)spec- In this theory, space and time in different frames ifying one-dimensional position x; a translation by X is of reference are treated as coordinate systems on a thus represented by      four-dimensional pseudo-Riemannian manifold (whose 10 00 tensors and = exp X . (1) mathematical background is described in X 1 10 manifolds [II.33]), which manifestly combines conser- When a physical system is invariant under a one- vation laws and continuous geometric symmetries of parameter group of transformations, the correspond- space and time. In special relativity (described in some ing generator plays a role in determining the associated detail in this article), this manifold is flat Minkowski conservation law. space-time, generalizing Euclidean space to include time in a physically natural way. In general relativity, 1 Mechanics in Euclidean Space described in detail in general relativity and cos- mology [IV.40], this manifold may be curved, depend- It is conventional in classical mechanics to define the ing in part on the distribution of matter and energy positions of a set of interacting particles in a vector according to einstein’s field equations [III.10]. space. However, we do not observe any unique origin In quantum physics, the description of a system in to the three-dimensional space we inhabit, which we terms of a complex vector in Hilbert space gives rise therefore take to be the Euclidean space E3; only rela- to new symmetries. An important example is the fact tive positions between different interacting subsystems

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108 II. Concepts

(i.e., positions relative to the common center of mass) (thereby capturing the forces as gradients of the poten- enter the equations of motion. The entire system may tial energy). Using the calculus of variations [IV.6], be translated in space without any effect on the phe- the functions qj (t) that satisfy the laws of mechanics nomena. are those that make the action stationary, and these Of course, external forces acting on the system may satisfy Lagrange’s equations of motion prevent this, such as a rubber ball in a linear gravity ∂L d ∂L − = 0,j= 1,...,n. (2) field. (In such situations the source of the force, such ∂qj dt ∂q˙j as the Earth as the source of gravity, is not considered The argument of the time derivative in this expression, part of the system.) The gravitational force may be rep- ∂L/∂q˙j , is called the canonical momentum pj for each resented by a potential V = gz for height z and gravita- j. The set of equations (2) involves the combination tional acceleration g; the ball’s mass m times the nega- of partial derivatives of the Lagrangian with respect to tive gradient, −m∇V , gives the downward force acting the coordinates and velocities, together with the total on the ball. The contours of V , given by z = const., derivative with respect to time. By the chain rule, this nevertheless have a symmetry: they are invariant to total derivative affects explicit time dependence in L translations of the horizontal coordinates x and y. and the implicit time dependence in each qj and q˙j . Since the gradient of the potential is proportional to Many of the conservation laws involving Lagrangians the gravitational force—which, by virtue of Newton’s involve such an interplay of explicit and implicit time law equals the rate of change of the particle’s linear dependence. momentum—the horizontal component of momentum Any transformation of the coordinates qj that does does not change and is therefore conserved even when not change the Lagrangian is a symmetry of the system. the particle bounces due to an impulsive, upward force If L does not have explicit dependence on a coordinate from the floor. The continuous, horizontal translational qj , then the first term in (2) vanishes: dpj /dt = 0, i.e., symmetry of the system therefore leads to conserva- the corresponding canonical momentum is conserved tion of linear momentum in the horizontal plane. In a in time. In the example from the last section of a par- similar argument employing Newton’s laws in cylindri- ticle in a linear gravitational field, the coordinates can cal polar coordinates, the invariance of the potential to be chosen to be Cartesian x,y,z, or cylindrical polars rotations about the z-axis leads to the conservation of r,φ,z; L is independent of x and y, leading to con- the vertical component a body’s angular momentum, servation of horizontal momentum, and also φ, lead- as observed for tops spinning frictionlessly. ing to conservation of angular momentum about the z-axis. The theorem is proved for symmetries of this type in classical mechanics [IV.19 §2.3]: if a system 2 Noether’s Theorem is homogeneous in space (translation invariant), then The Lagrangian framework for mechanics (classical linear momentum is conserved, and if it is isotropic mechanics [IV.19 §2]), which describes systems acting (independent of rotations, such as the Newtonian gravi- under forces defined by gradients of potentials (as in tational potential around a massive point particle exert- the previous section), is a natural mathematical setting ing a central force), then angular momentum is con- in which to explore the connection between a system’s served (equivalent to Kepler’s second law of planetary symmetries and its conservation laws. Here, a mechan- motion for gravity). ical system evolving in time t is described by n gener- Since it is the equations (2) that represent the phys- ical laws rather than the form of L or S, the system alized coordinates qj (t) and their time derivatives q˙j , may admit a more general kind of symmetry whose for j = 1,...,n, where the initial values qj (t0) at time transformation adds a time-dependent function to the t0 and final values qj (t1) at t1 are fixed. The action of the system is the functional Lagrangian L. If, under the transformation, the Lagran-  gian transforms L → L + dΛ/dt involving the total time t1 S[{qj }] = L({qj }, {q˙j },t)dt, derivative of some function Λ, the action transforms t 0 S→S+Λ(t1) − Λ(t0). Thus the transformed action where L({qj }, {q˙j },t)is the Lagrangian; this is a func- is still made stationary by functions satisfying (2), so tion of the coordinates, their corresponding velocities, transformations of this kind are symmetries of the sys- and maybe time, specified here by the total kinetic tem, which are also continuous if Λ also depends on a energy minus the total potential energy of the system continuous parameter s so that its time derivative is

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II.21. Invariants and Conservation Laws 109 zero when s = 0. It is not then difficult to see that the law, that force is proportional to acceleration. Accord- quantity ing to pre-Newtonian physics, forces were thought to be   n   ∂qj  ∂Λ proportional to velocity (as the effect of friction was not pj  −  , (3) ∂s = ∂s = fully appreciated), and it was not until Galileo’s thought j=1 s 0 s 0 experiments in friction-free environments that the pro- defined in terms of the generators of the transforma- portionality of force to acceleration was appreciated. In tion on each coordinate and the Lagrangian, is con- spite of Galilean invariance, problems involving circu- stant in time. This is Noether’s theorem for classical lar motion do in fact seem to require a privileged frame mechanics. of reference, called absolute space. One example due to An important example is when the Lagrangian has Newton himself is the problem of explaining, without no explicit dependence on time, ∂L/∂t = 0. In this absolute space, the meniscus formed by the surface of case, under an infinitesimal time translation t → t + δt, water in a spinning bucket; such problems are properly L → L+δtdL/dt, so here Λ is Lδt, with L evaluated at t, overcome only in general relativity. and δt plays the role of s. Under the same infinitesimal With Galilean relativity, absolute position is no longer → + transformation, qj qj δtq˙j , so the relevant con- defined: events occurring at the same position but at − served quantity (3) is j pj q˙j L, which is the Hamil- different times in one frame (such as a moving train car- tonian of the system, which is equal to the total energy riage) occur at different positions in other frames (such in many systems of interest. It is apparently a funda- as the frame of the train track). However, changes to the mental law of physics that the total energy in physical state of motion, i.e., accelerations, have physical conse- processes is conserved in time; energy can be in other quences and are related to forces. This is an example of forms such as electromagnetic, gravitational, or heat, a covariance principle, whose importance for physical as well as mechanical. Noether’s theorem states that theories was emphasized by Einstein. According to this the law of conservation of energy is equivalent to the principle, from the statement of physical laws in one fact that the physical laws of the system, characterized frame of reference (such as the laws of motion), one by their Lagrangian, do not change with time. can derive their statement in a different frame of refer- The vanishing of the action functional’s integrand ence from the application of the appropriate transfor- (i.e., the Lagrangian L) is equivalent to the existence mation rule between reference frames. The statement of a first integral for the system of Lagrange equa- in the new frame should have the same mathematical tions, which is interpreted in the mechanical setting form as in the previous frame, although quantities may as a constant of the motion of the system. In this not take the same values in different frames. sense, Noether’s theorem may be applied more gener- Transformations between different inertial frames ally in other physical situations described by function- are represented mathematically in a similar way to the als whose physical laws are given by the corresponding translations of (1); events are labeled by their positions Euler–Lagrange equations. In the case of the Lagran- in space and time, such as (t, x) in one frame and gian approach applied to fields (i.e., functions of space (t, x) in another moving at velocity v with respect to  = − and time), Noether’s theorem generalizes to give a con- the first. Since x x vt, the transformation from  10 tinuous density ρ (such as mass or charge density) (t, x) to (t, x ) is represented by the matrix −v 1 . and a flux vector J satisfying the continuity equation This Galilean transformation (or Galilean boost) differs ρ˙ +∇·J = 0 at every point in space and time. from (1) in that time t is here appended to the posi- tion vector, since the translation from the boost is time- 3 Galilean Relativity dependent. Galilean boosts in three spatial dimensions, together with regular translations and rotations, define Newton’s first law of motion can be paraphrased as “all the Galilean group. It can be shown that the Lagrangian inertial frames, traveling at uniform linear velocity with of a free particle follows directly from the covariance respect to each other, are equivalent for the formula- of the corresponding action under the Galilean group. tion of mechanics”—that is, without action of external Infinitesimal velocity boosts generate a Noetherian forces, a system will behave in the same way regardless symmetry on systems of particles interacting via forces of the motion of its center of mass. The behavior of a that depend only on the positions of the others. Con- mechanical system is therefore independent of its over- sider N point particles of mass mk and position rk such − = all velocity; this is a consequence of Newton’s second that V depends only on rk r- for k, - 1,...,N. Under

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110 II. Concepts

an infinitesimal boost by δv for each k, rk → rk + tδv, Rotations and Lorentz boosts, together with time reflec- r˙k → r˙k + δv, and tion and spatial inversion, define the Lorentz group; → + 1 | |2 + · the Poincaré group is the semidirect product group of L L 2 δv mk δv mkr˙k. k k the Lorentz group with space-time translations. Spe- The resulting Noetherian conserved quantity (3) is cial relativity is therefore also based on the principle of covariance, but with a different set of transformations = − C t mkr˙k mkrk. (4) between frames of reference. All of the familiar results k k of classical mechanics [IV.19] are recovered when This quantity is constant in time; its value is minus the the boost transformations are restricted to v  c. product of the system’s total mass with the position of In special relativity different observers measure dif- = ˙ = its center of mass when t 0. Thus, C t k mkr¨, ferent time intervals between events, as well as differ- that is, t times the sum of the forces on each of the N ent spatial separations. For a relativistic particle, differ- particles, which is zero by assumption. ent observers will disagree on the particle’s relativistic energy E and momentum p. Nevertheless, according to 4 Special Relativity the principle of special relativity, all observers agree on the quantity Historically, the physics of fields, such as electromag- E2 −|p|2c2 = m2c4, netic radiation described by maxwell’s equations [III.22], was developed much later than the mechanics so m is an invariant on which all observers agree, called of Galileo and Newton. The simplest wave equations, in the rest mass of the particle. In the frame in which the fact, suggest a different invariant space-time from the particle is at rest, E = mc2 arises as a special case. Galilean framework above, which was first investigated Otherwise, E = mγ(v)c2, so in special relativity the by Hendrik Lorentz and Albert Einstein at the turn of moving particle’s energy is explicitly related to the flow the twentieth century. of time in the particle’s rest frame with respect to that Consider the time-dependent wave equation ∇2ϕ − in the reference frame. c−2ϕ¨ = 0 for some scalar function ϕ(t, r) and ∇2 the The Lagrangian formalism can be generalized to spe- laplace operator [III.18]. If all observers agree on the cial relativity, with all observers agreeing on the form same value of the wave speed c (as proposed by Ein- of the Euler–Lagrange equations; instead of being equal stein for c the speed of light, and verified by experi- to its kinetic energy, the Lagrangian of a free particle is ments), this suggests that the transformation between −mc2/γ(v). moving inertial frames should be in terms of Lorentz The set of space-time events (ct, r) described by transformations (or Lorentz boosts)inthex-direction, special relativity is known as Minkowski space or      Minkowski space-time (discussed in tensors and man- ct 1 −v/c ct = γ(v) , (5) ifolds [II.33]). It is a flat four-dimensional manifold x −v/c 1 x with one time coordinate and three space coordinates   where (ct ,x ) denotes position in space and time (mul- (i.e., a manifold with a 3 + 1 pseudo-Euclidean metric), tiplied by the invariant speed c) in a frame moving and the inertial frames with perpendicular spatial axes along the x-axis at velocity v with respect to the frame are analogous to Cartesian coordinates in Euclidean with space-time coordinates (ct, x), and space. Time intervals and (simultaneous) spatial dis- − γ(v) = (1 − v2/c2) 1/2. tances between events are no longer separately invari- ant; only the space-time interval Requiring γ(v) to be finite, positive, and real-valued 2 =| |2 − 2 2 implies |v|

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II.21. Invariants and Conservation Laws 111 c, only events with s2 0 can be causally connected; follow curved space-time geodesics. The symmetries the events that may be affected by a given space-time of general relativistic space-time manifolds are much point are those within its future light cone. more complicated than those for Minkowski space but Being a manifold, the symmetries of Minkowski space are still formulated in terms of Killing vector fields that are easily characterized, and its continuous symme- generate transformations that keep equations invari- tries are generated by vector fields, called Killing vec- ant. tor fields: dynamics in the space-time manifold is In quantum physics (in a Galilean or special rela- not changed under infinitesimal pointwise translation tivistic setting), the ideas described in this article are along these fields. The symmetries are translations in very important in quantum field theory, required to four linearly independent space-time directions, rota- describe the quantum nature of electromagnetic and tion about three linearly independent spatial axes, and other fields, and systems involving many quantum par- Lorentz boosts in three linearly independent spatial ticles. In quantum field theory, all energies in the sys- directions (which are equivalent to rotations in space- tem are quantized, often around a minimum-energy time that mix the t-direction with a spatial direc- configuration. However, many field theories involve tion). There are therefore ten independent symmetry potentials whose minimum energy is not the most sym- transformations in Minkowski space-time (generating metric choice of origin; for instance, a “Mexican hat the Poincaré group), such that, for special relativis- potential” of the form |v|4 − 2a|v|2 for some field tic systems not experiencing external forces, relativis- vector v and constant a>0 is rotationally symmet- tic energy, momentum, angular momentum, and the = ric around√ v 0 but has a minimum for any v with analogue of C in (4) are conserved. |v|= a. A minimum-energy excitation of the sys- Both electromagnetic fields described by maxwell’s tem breaks this symmetry by choosing an appropriate equations [III.22] and relativistic quantum matter minimum energy v. Many phenomena in the quantum waves (strictly for a single particle) described by the theory of condensed matter (such as the Meissner effect dirac equation [III.9] can be expressed as Lagran- in superconductors) and fundamental particles (such gian field theories; the combination of these is called as the Higgs boson) arise from this type of symmetry classical field theory. As described above, Noether’s breaking. theorem relates continuous symmetries of these the- In classical mechanics, the time direction is privi- ories to continuity equations of relativistic 4-currents. leged (it is common to all observers), so its space- For instance, space-time translation symmetry ensures time structure cannot be phrased simply in terms of that the relativistic rank-2 stress–energy–momentum manifolds. Nevertheless, the Galilean group of trans- tensor, which describes the space-time flux of energy formations naturally has a fiber bundle structure, with (including matter), is divergence free. the base space given by the one-dimensional Euclidean line E1 describing time and the fiber being the spatial 5 Other Theories coordinates given by E3. Galilean transformations can naturally be built into Einstein’s theory of general relativity is a yet more gen- this fiber bundle structure, known as Newton–Cartan eral approach, which admits transformations between space-time, by defining the paths of free inertial parti- any coordinate systems on space-time (not simply inertial frames). This general formulation now applies cles (i.e., straight lines) in the connections of the bun- to an arbitrary coordinate system, such as rotating dle. This provides a useful mathematical framework coordinates (resolving the paradoxes implicit in New- to compare the physical laws and symmetries of New- ton’s bucket problem). This is formulated on a possi- tonian and relativity theories, and interesting connec- bly curved, pseudo-Riemannian manifold where local tions exist between general relativity and Newtonian neighborhoods of space-time events are equivalent to gravity in the Newton–Cartan formalism. Minkowski space and free particles follow geodesics on the manifold. The geometry of the manifold— Further Reading expressed by the Einstein curvature tensor—is pro- Lorentz, H. A., A. Einstein, H. Minkowski, and H. Weyl. 1952. portional, by einstein’s field equations [III.10], to The Principle of Relativity. New York: Dover. the stress–energy–momentum tensor, and gravitational Misner, C. W., K. S. Thorne, and J. A. Wheeler. 1973. Gravi- forces are fictitious forces as freely falling particles tation. New York: W. H. Freeman.

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112 II. Concepts

Neuenschwander, D. E. 2010. Emmy Noether’s Wonderful If every block Jk is 1 × 1 then J is diagonal and A is Theorem. Baltimore, MD: Johns Hopkins University Press. similar to a diagonal matrix; such matrices A are called Penrose, R. 2004. The Road to Reality (chapters 17–20 are diagonalizable. For example, real symmetric matrices particularly relevant). London: Random House. are diagonalizable—and moreover the eigenvalues are real and the matrix Z in the JCF can be taken to be orthogonal. A matrix that is not diagonalizable is defec- II.22 The Jordan Canonical Form tive; such matrices do not have a complete set of lin- Nicholas J. Higham early independent eigenvectors or, equivalently, their Jordan form has at least one block of dimension 2 or A canonical form for a class of matrices is a form of greater. matrix—usually chosen to be as simple as possible— To give a specific example, the matrix ⎡ ⎤ to which all members of the class can be reduced by ⎢ 311⎥ transformations of a specified kind. The Jordan canon- 1 ⎢ ⎥ A = ⎣−110⎦ (1) ical form (JCF) is associated with similarity transforma- 2 001 tions on n × n matrices. A similarity transformation of a matrix A is a transformation from A to X−1AX, where has a JCF with ⎡ ⎤ ⎡ ⎤ X is nonsingular. The JCF is the simplest form that can 0 1 1 1 00 ⎢ 2 ⎥ ⎢ 2 ⎥ be achieved by similarity transformations, in the sense ⎢ 1 ⎥ ⎢ ⎥ Z = ⎣−1 − 0⎦ ,J= ⎣ 0 11⎦ . that it is the closest to a diagonal matrix. 2 100 0 01 The JCF of a complex n × n matrix A can be writ- ten A = ZJZ−1, where Z is nonsingular and the Jordan As the partitioning of J indicates, there are two Jor- × 1 × dan blocks: a 1 1 block with eigenvalue 2 anda2 2 matrix J is a block-diagonal matrix 1 ⎡ ⎤ block with eigenvalue 1. The eigenvalue 2 of A has an J ⎢ 1 ⎥ associated eigenvector equal to the first column of Z. ⎢ ⎥ ⎢ J2 ⎥ ⎢ ⎥ For the double eigenvalue 1 there is only one linearly ⎢ . ⎥ ⎣ .. ⎦ independent eigenvector, namely the second column, z ,ofZ. The third column, z ,ofZ is a generalized Jp 2 3 eigenvector: it satisfies Az3 = z2 + z3. with diagonal blocks of the form ⎡ ⎤ The JCF provides complete information about the λ 1 ⎢ k ⎥ eigensystem. The geometric multiplicity of an eigen- ⎢ . ⎥ ⎢ . ⎥ value, defined as the number of associated linearly ⎢ λk . ⎥ J = J (λ ) = ⎢ ⎥ . independent eigenvectors, is the number of Jordan k k k ⎢ .. ⎥ ⎣ . 1 ⎦ blocks in which that eigenvalue appears. The algebraic λk multiplicity of an eigenvalue, defined as its multiplic- = Here, blanks denote zero blocks or zero entries. The ity as a zero of the characteristic polynomial q(t) − matrix J is unique up to permutation of the diago- det(tI A), is the number of copies of the eigenvalue among all the Jordan blocks. For the matrix (1) above, nal blocks, but Z is not. Each λk is an eigenvalue of A and may appear in more than one Jordan block. All the geometric multiplicity of the eigenvalue 1 is 1 and the algebraic multiplicity is 2, while the eigenvalue 1 the eigenvalues [I.2 §20] of the Jordan block Jk are 2 has geometric and algebraic multiplicities both equal equal to λk. By definition, an eigenvector of Jk is a to 1. nonzero vector x satisfying Jkx = λkx, and all such x are nonzero multiples of the vector x = [10 ··· 0]T. The minimal polynomial of a matrix is the unique monic polynomial ψ of lowest degree such that ψ(A) = Therefore Jk has only one linearly independent eigen- vector. Expand x to a vector x˜ with n components by 0. The degree of ψ is certainly no larger than n cayley–hamilton theorem padding it with zeros in positions corresponding to because the [IV.10 §5.3] states that q(A) = 0. The minimal polynomial of an each of the other Jordan blocks Ji, i = k. The vec- × − m tor x˜ has a single 1, in the r th component, say. A m m Jordan block Jk(λk) is (t λk) . The minimal corresponding eigenvector of A is Zx˜, since A(Zx)˜ = polynomial of A is therefore given by −1 s ZJZ (Zx)˜ = ZJx˜ = λkZx˜; this eigenvector is the r th ψ(t) = (t − λ )mi , column of Z. i i=1

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