DOCSLIB.ORG

Topology and Bifurcations in Hamiltonian Coupled Cell Systems

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Topology and Bifurcations in Hamiltonian Coupled Cell Systems To appear in Dynamical Systems: An International Journal Vol. 00, No. 00, Month 20XX, 1–22 Topology and Bifurcations in Hamiltonian Coupled Cell Systems a b c B.S. Chan and P.L. Buono and A. Palacios ∗ aDepartment of Mathematics,San Diego State University, San Diego, CA 92182; bFaculty of Science, University of Ontario Institute of Technology, 2000 Simcoe St N, Oshawa, ON L1H 7K4, Canada; cDepartment of Mathematics,San Diego State University, San Diego, CA 92182 (v5.0 released February 2015) The coupled cell formalism is a systematic way to represent and study coupled nonlinear differential equations using directed graphs. In this work, we focus on coupled cell systems in which individual cells are also Hamiltonian. We show that some coupled cell systems do not admit Hamiltonian vector fields because the associated directed graphs are incompatible. In broad terms, we prove that only sys- tems with bidirectionally coupled digraphs can be Hamiltonian. Aside from the topological criteria, we also study the linear theory of regular Hamiltonian coupled cell systems, i.e., systems with only one type of node and one type of coupling. We show that the eigenspace at a codimension one bifurcation from a synchronous equilibrium of a regular Hamiltonian network can be expressed in terms of the eigenspaces of the adjacency matrix of the associated directed graph. We then prove results on steady-state bifurca- tions and a version of the Hamiltonian Hopf theorem. Keywords: Hamiltonian systems; coupled cells; bifurcations; nonlinear oscillators 37C80; 37G40; 34C14; 37K05 1. Introduction The study of coupled systems of differential equations, also known as coupled cell systems, received much attention recently with various theories and approaches being developed con- currently [1–5]. The groupoid formalism approach to studying coupled cell systems developed by M. Golubitsky, I. Stewart, A. Dias and many other collaborators has shown that such types of systems exhibit generically bifurcation phenomena that are not observed in systems without this structure [6–11]; e.g. patterns of synchrony solutions, nilpotent bifurcations, including more than (1/2)th-power growth at Hopf bifurcation. A coupled cell system where all cells in the net- work are identical is called a homogeneous network and it is regular if all couplings are of the same type and each cell receives the same number of inputs. In this paper, we study regular coupled cell systems in which individual cells are also Hamilto- nian systems. Our main results are the following. We begin by considering a system of coupled Hamiltonian equations with an equilibrium at the origin and show a necessary and sufficient condition for the linear part at the origin to be a Hamiltonian matrix. We link this result with the structure of the adjacency matrix of the coupled cell network. Then we use this result to give a necessary condition on the digraph of a nonlinear Hamiltonian coupled cell system and also a necessary and sufficient condition for linearly coupled Hamiltonian cells. The second part of the paper is concerned with extending results of Golubitsky and Lauterbach [8] on the critical ∗Corresponding author. Email: [email protected] eigenspaces for codimension one families of regular coupled cell systems to the Hamiltonian case. In the non-Hamiltonian case, for one-parameter families of linear systems, generically, bi- furcations occur as simple eigenvalues cross transversally the imaginary axis. On the other hand, in the Hamiltonian case, eigenvalues come in quadruplets λ, λ, λ,¯ λ¯ [12] and so bifurcations occur from collisions of eigenvalues on the imaginary axis, thus− leading− generically to having eigenvalues with non-semisimple eigenspaces. We show that for zero eigenvalues of multiplicity two and purely imaginary eigenvalues in 1 : 1 resonance, the generalized eigenspace can be expressed in terms of the eigenspace of the adjacency− matrix of the graph. We obtain bifurcation results leading to steady-states in synchrony subspaces (as a generalization of the Hamiltonian equivariant branching lemma [13]) and a generalization of the Hamiltonian Hopf bifurcation to synchrony subspaces. Coupled cell systems with Hamiltonian structure arise, for instance, in the context of analyz- ing the bifurcation structure in symmetrically coupled ring networks of gyroscopes [14] and of energy harvesters [15]. In these two cases, the Hamiltonian structure of each cell is obtained by setting the linear damping term to zero. This is a reasonable assumption as the damping term coefficient is several orders of magnitude smaller than the other parameters. It is shown in [14] that unidirectionally coupled ring networks of linear Hamiltonian systems do not preserve the Hamiltonian structure while the bidirectionally coupled rings do keep the structure. In the en- ergy harvester case, the network is all-to-all coupled with SN permutation symmetry group and the Hamiltonian structure is also preserved. These finding prompted us to investigate the gener- alization to arbitrary coupled networks of Hamiltonian cells presented in this paper. Many systems of coupled Hamiltonian systems have been investigated over the years, al- though not necessarily using the graph theoretic formalism described in this paper. For instance, N-body problems in the form of “kinetic+potential” with the kinetic part describing the free motion of each body can be thought of coupled Hamiltonian systems with the potential func- tion acting as the coupling term via the configuration variables. In the case of the Newtonian N-body problem, the potential acts as an all-to-all coupling term (see [16]), while in models of molecules, the potential energy is the coupling term describing the electronic binding between the atoms, see [17]. Other examples of coupled Hamiltonian systems are the Fermi-Pasta-Ulam (FPU) chains [18] modelling an infinite number of particles coupled via a potential function de- pending only on the positions. The case of finite number of particles is studied given boundary conditions; for instance, fixed endpoints leading to a Dn symmetric coupled cell system [19]. Fi- nally, we mention the study of the free motion of coupled rigid bodies such as described in [20] where the Lagrangian is given as the kinetic energy of each body, with coupling via a hinge constraint. However, in this case the dynamical equations obtained via symplectic reduction do not preserve the coupled cell structure as described in this paper. Recently, Manoel and Roberts [21] have studied a related problem to the one studied here, that is, whether a network can be regarded as a gradient system. They also determine the requirement that the digraph must be symmetric and characterize the form of admissible functions defining a gradient coupled cell system. Their main results are about regular graphs and they show neces- sary and sufficient conditions for a point to be a critical point of the admissible function in terms of the coupling function. They also mention how their results apply to the Kuramoto model and the Antiferromagnetic XY model. The paper is organized as follows. In Section 2, we introduce a graph theoretic definition of coupled cell systems and establish linear and nonlinear criteria for the coupled cell system to be Hamiltonian, given that each cell is Hamiltonian. Section 3 discusses generalized eigenspaces properties of the Hamiltonian coupled systems in terms of the eigenspaces of the eigenvalues of the adjacency matrix of the graph. This section specializes the results of [8] to the case of Hamil- tonian coupled cell systems. In particular, we obtain the structure of the generalized eigenspace of the Jacobian to be isomorphic to copies of the eigenspace of a given eigenvalue of the adja- 2 cency matrix. Section 4 presents the steady-state bifurcation and Hamiltonian Hopf bifurcations results when restricted to synchrony subspaces. The final section presents a summary of our results and a short discussion on future work. 2. Hamiltonian coupled cell systems In a coupled cell system, each cell is a system of differential equations with phase space variable Rki xi , for i 1,..., n . Suppose that cell i receives input from cells j1,..., jmi 1,..., n , then∈ the dynamics∈ { of the }ith component is ∈ { } dxi = fi(xi, x j ,..., x j ). dt 1 mi Another feature of this formalism is that a coupled cell system can be represented graphically using so-called directed graphs. Definition 2.1: A directed graph (or digraph) G consists of a vertex set V(G) and an arc set E(G), where an arc is an ordered pair of distinct vertices. As an example, the graph at the top of Figure 1 has vertex set V = v , v and arc set E = { 1 2} e1, e2 . In this situation, vertices v1 and v2 represent the internal dynamics of the two cells. Similarly,{ } arcs e = 2, 1 and e = 1, 2 represent the coupling dynamics between the two 1 { } 2 { } vertices. Figure 1 (bottom) has vertex set V = v1, v2, v3 and arc set E = e1, e2, e3, e4 with e = 2, 1 , e = 1, 2 , e = 3, 1 and e = 2, 3 .{ } { } 1 { } 2 { } 3 { } 4 { } e1 v1 v2 e2 e3 e1 e4 v1 v2 v3 e2 Figure 1.: Examples of digraphs representing coupled cell systems. Given this setup, we may view the system as dxi = gi(xi) + hi(x j ,..., x j ), dt 1 mi where gi represents the dynamics pertaining to cell i and hi is the function of the inputs into cell i. 3 = If we assume each cell dynamics has an equilibrium solution gi(xi∗) 0, then we can translate the equilibrium (x1∗,..., xn∗) to the origin. Without loss of generality, we can assume that the system has an equilibrium at the origin. Then, at the linear level, the matrices for internal and coupling dynamics can be written as ∂gi ∂hi = Qi and = Rij, (1) ∂x = ∂x i x 0 j x=0 where x = (x ,..., x )T , Q Rki ki , and R Rki k j .
Load more

Recommended publications
  • Lecture 6 — Generalized Eigenspaces & Generalized Weight
    18.745 Introduction to Lie Algebras September 28, 2010 Lecture 6 | Generalized Eigenspaces & Generalized Weight Spaces Prof. Victor Kac Scribe: Andrew Geng and Wenzhe Wei Definition 6.1. Let A be a linear operator on a vector space V over field F and let λ 2 F, then the subspace N Vλ = fv j (A − λI) v = 0 for some positive integer Ng is called a generalized eigenspace of A with eigenvalue λ. Note that the eigenspace of A with eigenvalue λ is a subspace of Vλ. Example 6.1. A is a nilpotent operator if and only if V = V0. Proposition 6.1. Let A be a linear operator on a finite dimensional vector space V over an alge- braically closed field F, and let λ1; :::; λs be all eigenvalues of A, n1; n2; :::; ns be their multiplicities. Then one has the generalized eigenspace decomposition: s M V = Vλi where dim Vλi = ni i=1 Proof. By the Jordan normal form of A in some basis e1; e2; :::en. Its matrix is of the following form: 0 1 Jλ1 B Jλ C A = B 2 C B .. C @ . A ; Jλn where Jλi is an ni × ni matrix with λi on the diagonal, 0 or 1 in each entry just above the diagonal, and 0 everywhere else. Let Vλ1 = spanfe1; e2; :::; en1 g;Vλ2 = spanfen1+1; :::; en1+n2 g; :::; so that Jλi acts on Vλi . i.e. Vλi are A-invariant and Aj = λ I + N , N nilpotent. Vλi i ni i i From the above discussion, we obtain the following decomposition of the operator A, called the classical Jordan decomposition A = As + An where As is the operator which in the basis above is the diagonal part of A, and An is the rest (An = A − As).
    [Show full text]
  • Calculus and Differential Equations II
    Calculus and Differential Equations II MATH 250 B Linear systems of differential equations Linear systems of differential equations Calculus and Differential Equations II Second order autonomous linear systems We are mostly interested with2 × 2 first order autonomous systems of the form x0 = a x + b y y 0 = c x + d y where x and y are functions of t and a, b, c, and d are real constants. Such a system may be re-written in matrix form as d x x a b = M ; M = : dt y y c d The purpose of this section is to classify the dynamics of the solutions of the above system, in terms of the properties of the matrix M. Linear systems of differential equations Calculus and Differential Equations II Existence and uniqueness (general statement) Consider a linear system of the form dY = M(t)Y + F (t); dt where Y and F (t) are n × 1 column vectors, and M(t) is an n × n matrix whose entries may depend on t. Existence and uniqueness theorem: If the entries of the matrix M(t) and of the vector F (t) are continuous on some open interval I containing t0, then the initial value problem dY = M(t)Y + F (t); Y (t ) = Y dt 0 0 has a unique solution on I . In particular, this means that trajectories in the phase space do not cross. Linear systems of differential equations Calculus and Differential Equations II General solution The general solution to Y 0 = M(t)Y + F (t) reads Y (t) = C1 Y1(t) + C2 Y2(t) + ··· + Cn Yn(t) + Yp(t); = U(t) C + Yp(t); where 0 Yp(t) is a particular solution to Y = M(t)Y + F (t).
    [Show full text]
  • SUPPLEMENT on EIGENVALUES and EIGENVECTORS We Give
    SUPPLEMENT ON EIGENVALUES AND EIGENVECTORS We give some extra material on repeated eigenvalues and complex eigenvalues. 1. REPEATED EIGENVALUES AND GENERALIZED EIGENVECTORS For repeated eigenvalues, it is not always the case that there are enough eigenvectors. Let A be an n × n real matrix, with characteristic polynomial m1 mk pA(λ) = (λ1 − λ) ··· (λk − λ) with λ j 6= λ` for j 6= `. Use the following notation for the eigenspace, E(λ j ) = {v : (A − λ j I)v = 0 }. We also define the generalized eigenspace for the eigenvalue λ j by gen m j E (λ j ) = {w : (A − λ j I) w = 0 }, where m j is the multiplicity of the eigenvalue. A vector in E(λ j ) is called a generalized eigenvector. The following is a extension of theorem 7 in the book. 0 m1 mk Theorem (7 ). Let A be an n × n matrix with characteristic polynomial pA(λ) = (λ1 − λ) ··· (λk − λ) , where λ j 6= λ` for j 6= `. Then, the following hold. gen (a) dim(E(λ j )) ≤ m j and dim(E (λ j )) = m j for 1 ≤ j ≤ k. If λ j is complex, then these dimensions are as subspaces of Cn. gen n (b) If B j is a basis for E (λ j ) for 1 ≤ j ≤ k, then B1 ∪ · · · ∪ Bk is a basis of C , i.e., there is always a basis of generalized eigenvectors for all the eigenvalues. If the eigenvalues are all real all the vectors are real, then this gives a basis of Rn. (c) Assume A is a real matrix and all its eigenvalues are real.
    [Show full text]
  • Math 2280 - Lecture 23
    Math 2280 - Lecture 23 Dylan Zwick Fall 2013 In our last lecture we dealt with solutions to the system: ′ x = Ax where A is an n × n matrix with n distinct eigenvalues. As promised, today we will deal with the question of what happens if we have less than n distinct eigenvalues, which is what happens if any of the roots of the characteristic polynomial are repeated. This lecture corresponds with section 5.4 of the textbook, and the as- signed problems from that section are: Section 5.4 - 1, 8, 15, 25, 33 The Case of an Order 2 Root Let’s start with the case an an order 2 root.1 So, our eigenvalue equation has a repeated root, λ, of multiplicity 2. There are two ways this can go. The first possibility is that we have two distinct (linearly independent) eigenvectors associated with the eigen- value λ. In this case, all is good, and we just use these two eigenvectors to create two distinct solutions. 1Admittedly, not one of Sherlock Holmes’s more popular mysteries. 1 Example - Find a general solution to the system: 9 4 0 ′ x = −6 −1 0 x 6 4 3 Solution - The characteristic equation of the matrix A is: |A − λI| = (5 − λ)(3 − λ)2. So, A has the distinct eigenvalue λ1 = 5 and the repeated eigenvalue λ2 =3 of multiplicity 2. For the eigenvalue λ1 =5 the eigenvector equation is: 4 4 0 a 0 (A − 5I)v = −6 −6 0 b = 0 6 4 −2 c 0 which has as an eigenvector 1 v1 = −1 .
    [Show full text]
  • 23. Eigenvalues and Eigenvectors
    23. Eigenvalues and Eigenvectors 11/17/20 Eigenvalues and eigenvectors have a variety of uses. They allow us to solve linear difference and differential equations. For many non-linear equations, they inform us about the long-run behavior of the system. They are also useful for defining functions of matrices. 23.1 Eigenvalues We start with eigenvalues. Eigenvalues and Spectrum. Let A be an m m matrix. An eigenvalue (characteristic value, proper value) of A is a number λ so that× A λI is singular. The spectrum of A, σ(A)= {eigenvalues of A}.− Sometimes it’s possible to find eigenvalues by inspection of the matrix. ◮ Example 23.1.1: Some Eigenvalues. Suppose 1 0 2 1 A = , B = . 0 2 1 2 Here it is pretty obvious that subtracting either I or 2I from A yields a singular matrix. As for matrix B, notice that subtracting I leaves us with two identical columns (and rows), so 1 is an eigenvalue. Less obvious is the fact that subtracting 3I leaves us with linearly independent columns (and rows), so 3 is also an eigenvalue. We’ll see in a moment that 2 2 matrices have at most two eigenvalues, so we have determined the spectrum of each:× σ(A)= {1, 2} and σ(B)= {1, 3}. ◭ 2 MATH METHODS 23.2 Finding Eigenvalues: 2 2 × We have several ways to determine whether a matrix is singular. One method is to check the determinant. It is zero if and only the matrix is singular. That means we can find the eigenvalues by solving the equation det(A λI)=0.
    [Show full text]
  • Math 4571 – Lecture 25
    Math 4571 { Lecture 25 Math 4571 (Advanced Linear Algebra) Lecture #25 Generalized Eigenvectors: Jordan-Block Matrices and the Jordan Canonical Form Generalized Eigenvectors Generalized Eigenspaces and the Spectral Decomposition This material represents x4.3.1 from the course notes. Math 4571 { Lecture 25 Jordan Canonical Form, I In the last lecture, we discussed diagonalizability and showed that there exist matrices that are not conjugate to any diagonal matrix. For computational purposes, however, we might still like to know what the simplest form to which a non-diagonalizable matrix is similar. The answer is given by what is called the Jordan canonical form, which we now describe. Important Note: The proofs of the results in this lecture are fairly technical, and it is NOT necessary to follow all of the details. The important part is to understand what the theorems say. Math 4571 { Lecture 25 Jordan Canonical Form, II Definition The n × n Jordan block with eigenvalue λ is the n × n matrix J having λs on the diagonal, 1s directly above the diagonal, and zeroes elsewhere. Here are the general Jordan block matrices of sizes 2, 3, 4, and 5: 2 λ 1 0 0 0 3 2 λ 1 0 0 3 2 λ 1 0 3 0 λ 1 0 0 λ 1 0 λ 1 0 6 7 , 0 λ 1 , 6 7, 6 0 0 λ 1 0 7. 0 λ 4 5 6 0 0 λ 1 7 6 7 0 0 λ 4 5 6 0 0 0 λ 1 7 0 0 0 λ 4 5 0 0 0 0 λ Math 4571 { Lecture 25 Jordan Canonical Form, III Definition A matrix is in Jordan canonical form if it is a block-diagonal matrix 2 3 J1 6 J2 7 6 7, where each J ; ··· ; J is a Jordan block 6 .
    [Show full text]
  • JORDAN FORM Contents 1. Eigenvectors and Generalized
    JORDAN FORM BENEDICT MORRISSEY Contents 1. Eigenvectors and Generalized Eigenvectors 1 2. Diagonalization 2 2.1. Extended Example – writing a matrix in Jordan Form 4 These are supplementary notes for parts of Lectures 8,9 and 10 of Math 312 Summer II 2019. 1. Eigenvectors and Generalized Eigenvectors Let T : V → V be a linear function, from a vector space V to itself. Definition 1.1. An eigenvector of T (corresponding to the eigenvalue λ)is a vector (which we generally require to be non zero) ~v such that T~v = λ~v for some scalar λ. We call a λ such that there is an eigenvector, an eigenvalue of T . We sometimes call this a λ-eigenvector. Equivalently ~v ∈ Ker(T − λId) (where Id is the identity matrix). We recall that the kernel of a linear transformation is zero precisely when the determinant is zero. Hence the eigenvalues of a linear transformation T are precisely the values of λ that are roots of the polynomial p(λ) := det(A − λId). We call p(λ) the characteristic polynomial of the linear transformation T . Definition 1.2. A generalized eigenvector of T (corresponding to the eigenvalue λ) is a vector ~v, such that there is some positive integer k such that (T − λId)k~v = 0 We say that a generalized eigenvector of T is of rank k if (T − λId)k~v = 0 and (T − λId)k−1~v 6= 0. We sometimes call this a generalized λ-eigenvector. Definition 1.3. For λ an eigenvalue of T we let Vλ ⊂ V be the vector subspace of V consisting of generalized eigenvectors of T corresponding to the eigenvalue λ (here we include the zero vector as a generalized eigenvector for the purposes of this definition).
    [Show full text]
  • Linear Algebra Notes
    LINEAR ALGEBRA OUTLINE UNIVERSITY OF ARIZONA INTEGRATION WORKSHOP, AUGUST 2015 These notes are intended only as a rough summary of the background in linear algebra expected of incoming graduate students. With brevity as our guiding principle, we include virtually no proofs of the results stated in the text below. The reader is encouraged to fill in as many of the arguments and details as possible. 1. Lecture 1 Throughout these notes, we fix a field k. 1.1. First definitions and properties. Definition 1.1.1. A vector space over k is a set V equipped with maps + : V × V ! V (addition) and · : k × V ! V (scalar multiplication) which satisfy: (1) The pair (V; +) forms an abelian group. That is: (a) + is commutative: v + w = w + v for all v; w 2 V . (b) + is associative: u + (v + w) = (u + v) + w for all u; v; w 2 V . (c) There exists an additive identity 0 2 V such that 0 + v = v for all v 2 V . (d) There exist additive inverses: for all v 2 V , there exists −v 2 V such that v + (−v) = 0. (2) Scalar multiplication gives an action of the group k× on V . That is: (a) a · (b · v) = (ab) · v for all a; b 2 k and all v 2 V . (b)1 · v = v for all v 2 V . (3) Scalar multiplication distributes over addition. That is: (a) a · (v + w) = a · v + a · w for all a 2 k and all v; w 2 V . (b)( a + b) · v = a · v + b · v for all a; b 2 k and all v 2 V .
    [Show full text]
  • On the Perron Roots of Principal Submatrices of Co-Order One of Irreducible Nonnegative Matricesୋ S.V
    Linear Algebra and its Applications 361 (2003) 257–277 www.elsevier.com/locate/laa On the Perron roots of principal submatrices of co-order one of irreducible nonnegative matricesୋ S.V. Savchenko L.D. Landau Institute for Theoretical Physics, Russian Academy of Sciences, Kosygina str. 2, Moscow 117334, Russia Received 15 September 2001; accepted 5 July 2002 Submitted by R.A. Brualdi Abstract Let A be an irreducible nonnegative matrix and λ(A) be the Perron root (spectral radius) of A. Denote by λmin(A) the minimum of the Perron roots of all the principal submatrices of co- order one. It is well known that the interval (λmin(A), λ(A)) does not contain any eigenvalues of A. Consider any principal submatrix A − v of co-order one whose Perron root is equal to λmin(A). We show that the Jordan structure of λmin(A) as an eigenvalue of A is obtained from that of the Perron root of A − v as follows: one largest Jordan block disappears and the others remain the same. So, if only one Jordan block corresponds to the Perron root of the submatrix, then λmin(A) is not an eigenvalue of A. By Schneider’s theorem, this holds if and only if there is a Hamiltonian chain in the singular digraph of A − v. In the general case the Jordan structure for the Perron root of the submatrix A − v and therefore that for the eigenvalue λmin(A) of A can be arbitrary. But if the Perron root λ(A − w) of a principal submatrix A − w of co-order one is strictly greater than λmin(A), then λ(A − w) is a simple eigenvalue of A − w.
    [Show full text]
  • Distinct, Real Eigenvalues
    Solve x0 = Ax From Section 7.1, we already know how to solve a system of two first order differential equations using the methods of Chapter 3 (after converting to a single second order equation). In these notes, we want to connect the solution to the system with the eigenvalues and eigenvectors of the coefficient matrix A. Given 0 " 0 # " #" # x1 = ax1 + bx2 x1 a b x1 0 0 or 0 = or x = Ax x2 = cx1 + dx2 x2 c d x2 Inspired by Chapter 3, we use the ansatz: " # " rt # rt rt v1 e v1 x = e v = e = rt v2 e v2 and substitute it into the differential equation. We will see that r is actually an eigenvalue of A, and v the corresponding eigenvector. First we compute x0, then Ax, then we set them equal to each other: " rt # 0 re v1 rt x = rt = re v re v2 and: Ax = Aertv = ertAv Set these equal to each other: ertAv = rertv We can divide by ert since it is never zero, and we get: Av = rv so that r is an eigenvalue of A (we will now stick with the λ notation), and v is the corre- sponding eigenvector. This is not the whole story, however. Just as we did in Chapter 3, we will need to find a fundamental set of solutions for our system. And, just as in Chapter 3, we will see that it takes two linearly independent solutions to form that fundamental set. We will have three cases classified by the eigenvalues (and in Chapter 3 by the roots to the characteristic equation): (i) Two distinct real eigenvalues, (ii) Complex conjugate eigenvalue, and (iii) One eigenvalue, one eigenvector.
    [Show full text]
  • Generalized Eigenvector - Wikipedia
    11/24/2018 Generalized eigenvector - Wikipedia Generalized eigenvector In linear algebra, a generalized eigenvector of an n × n matrix is a vector which satisfies certain criteria which are more relaxed than those for an (ordinary) eigenvector.[1] Let be an n-dimensional vector space; let be a linear map in L(V), the set of all linear maps from into itself; and let be the matrix representation of with respect to some ordered basis. There may not always exist a full set of n linearly independent eigenvectors of that form a complete basis for . That is, the matrix may not be diagonalizable.[2][3] This happens when the algebraic multiplicity of at least one eigenvalue is greater than its geometric multiplicity (the nullity of the matrix , or the dimension of its nullspace). In this case, is called a defective eigenvalue and is called a defective matrix.[4] A generalized eigenvector corresponding to , together with the matrix generate a Jordan chain of linearly independent generalized eigenvectors which form a basis for an invariant subspace of .[5][6][7] Using generalized eigenvectors, a set of linearly independent eigenvectors of can be extended, if necessary, to a complete basis for .[8] This basis can be used to determine an "almost diagonal matrix" in Jordan normal form, similar to , which is useful in computing certain matrix functions of .[9] The matrix is also useful in solving the system of linear differential equations where need not be diagonalizable.[10][11] Contents Overview and definition Examples Example 1 Example 2 Jordan chains
    [Show full text]
  • Eigenvectors from Eigenvalues: a Survey of a Basic Identity in Linear Algebra
    EIGENVECTORS FROM EIGENVALUES: A SURVEY OF A BASIC IDENTITY IN LINEAR ALGEBRA PETER B. DENTON, STEPHEN J. PARKE, TERENCE TAO, AND XINING ZHANG Abstract. If A is an n×n Hermitian matrix with eigenvalues λ1(A); : : : ; λn(A) th and i; j = 1; : : : ; n, then the j component vi;j of a unit eigenvector vi associ- ated to the eigenvalue λi(A) is related to the eigenvalues λ1(Mj ); : : : ; λn−1(Mj ) th of the minor Mj of A formed by removing the j row and column by the for- mula n n−1 2 Y Y jvi;j j (λi(A) − λk(A)) = (λi(A) − λk(Mj )) : k=1;k6=i k=1 We refer to this identity as the eigenvector-eigenvalue identity and show how this identity can also be used to extract the relative phases between the com- ponents of any given eigenvector. Despite the simple nature of this identity and the extremely mature state of development of linear algebra, this identity was not widely known until very recently. In this survey we describe the many times that this identity, or variants thereof, have been discovered and redis- covered in the literature (with the earliest precursor we know of appearing in 1834). We also provide a number of proofs and generalizations of the identity. 1. Introduction If A is an n×n Hermitian matrix, we denote its n real eigenvalues by λ1(A); : : : ; λn(A). The ordering of the eigenvalues will not be of importance in this survey, but for sake of concreteness let us adopt the convention of non-decreasing eigenvalues: λ1(A) ≤ · · · ≤ λn(A): th If 1 ≤ j ≤ n, let Mj denote the n−1×n−1 minor formed from A by deleting the j row and column from A.
    [Show full text]