115 Nonlinear Eigenvalue Problems 115.1 Basic Properties :::::::::::::::::::::::::::::::::::::::: 115-2 115.2 Analytic matrix functions ::::::::::::::::::::::::::::: 115-3 115.3 Variational Characterization of Eigenvalues :::::::: 115-7 115.4 General Rayleigh Functionals :::::::::::::::::::::::: 115-9 115.5 Methods for dense eigenvalue problems ::::::::::::: 115-10 115.6 Iterative projection methods:::::::::::::::::::::::::: 115-13 115.7 Methods using invariant pairs :::::::::::::::::::::::: 115-17 Heinrich Voss 115.8 The infinite Arnoldi method :::::::::::::::::::::::::: 115-20 Hamburg University of Technology References:::::::::::::::::::::::::::::::::::::::::::::::::::::: 115-22 This chapter considers the nonlinear eigenvalue problem to find a parameter λ such that the linear system T (λ)x = 0 (115.1) has a nontrivial solution x, where T (·): D ! Cn×n is a family of matrices depending on a complex parameter λ 2 D. It generalizes the linear eigenvalue problem Ax = λx, A 2 Cn×n, where T (λ) = λI − A, and the generalized linear eigenvalue problem where T (λ) = λB − A, A; B 2 Cn×n. Nonlinear eigenvalue problems T (λ)x = 0 arise in a variety of applications in science and engineering, such as the dynamic analysis of structures, vibrations of fluid{solid structures, the electronic behavior of quantum dots, and delay eigenvalue problems, to name just a few. Due to its wide range of applications, the quadratic eigenvalue problem T (λ)x = λ2Mx + λCx + Kx = 0 is of particular interest, but also polynomial, rational and more general eigenvalue problems appear. A standard approach for investigating or numerically solving polynomial eigenvalue problems is linearization where the original problem is transformed into a generalized linear eigenvalue problem with the same spectrum. Details on linearization and structure preservation are discussed in Chapter 102, Matrix Polynomials. This chapter is concerned with the general nonlinear eigenvalue problem which in general can not be linearized. Unlike for linear and polynomial eigenvalue problems there may exist infinitely many eigenvalues. In practice, however, one is usually interested only in a few eigenvalues close to a target value or a line in the complex plane. If T is linear then T (λ) = T (0)+λT 0(0) has the form of a generalized eigenvalue problem, and in the general case linerization gives the approximation T (λ) = T (0) + λT 0(0) + O(λ2), which is again a generalized linear eigenvalue problem. Hence, it is not surprising, that the (elementwise) derivative T 0(λ) of T (λ) plays an important role in the analysis of nonlinear eigenvalue problems. We tacitly assume in the whole chapter that whenever a derivative T 0(λ^) appears, T is analytic in a neighborhood of λ^ or in the real case T : D ! Rn×n, D ⊂ R that T is differentiable in a neighborhood of λ^. k · k always denotes the Euclidean and spectral norm, respectively, and we use the notation [x; y] := [xT ; yT ]T for column vectors. 115-1 115-2 Handbook of Linear Algebra 115.1 Basic Properties This section presents basic properties of the nonlinear eigenvalue problem (115.1) Definitions: As for a linear eigenvalue problem, λ^ 2 D is called an eigenvalue of T (·) if T (λ^)x = 0 has a nontrivial solution x^ 6= 0. Then x^ is called a corresponding eigenvector or right eigenvector, and (λ,^ x^) is called eigenpair of T (·). Any nontrivial solution y^ 6= 0 of the adjoint equation T (λ^)∗y = 0 is called left eigenvector of T (·) and the vector-scalar-vector triplet (y^; λ,^ x^) is called eigentriplet of T (·). The eigenvalue problem (115.1) is regular if det T (λ) 6≡ 0, and otherwise it is called singular. The spectrum σ(T (·)) of T (·) is the set of all eigenvalues of T (·). ` ^ d An eigenvalue λ of T (·) has algebraic multiplicity k if ` det(T (λ)) = 0 for ` = 0; : : : ; k− dλ λ=λ^ dk 1 and k det(T (λ)) 6= 0. dλ λ=λ^ An eigenvalue λ^ is simple if its algebraic multiplicity is one. The geometric multiplicity of an eigenvalue λ^ is the dimension of the kernel ker(T (λ^)) of T (λ^). An eigenvalue λ^ is called semi-simple if its algebraic and geometric mutiplicity coincide. n×n T T (·): J ! R is real symmetric if T (λ) = T (λ) for every λ 2 J ⊂ R. n×n T T (·): D ! C is complex symmetric if T (λ) = T (λ) for every λ 2 D. n×n ∗ T (·): D ! C is Hermitian if D is symmetric with respect to the real line and T (λ) = T (λ¯) for every λ 2 D. Facts: 1. For A 2 Cn×n and T (λ) = λI − A, the terms eigenvalue, (left and right) eigen- vector, eigenpair, eigentriplet, spectrum, algebraic and geometric multiplicity and semi-simple have their standard meaning. 2. For linear eigenvalue problems, { eigenvectors corresponding to distinct eigenvalues are linearly independent, which is not the case for nonlinear eigenvalue problems (cf. Example 1). { left and right eigenvectors corresponding to distinct eigenvalues are orthogonal, which does not hold for nonlinear eigenproblems (cf. Example 2). { the algebraic multiplicities of eigenvalues sum up to the dimension of the prob- lem, whereas for nonlinear problems there may exist an infinite number of eigen- values (cf. Example 2) and an eigenvalue may have any algebraic multiplicity (cf. Example 3). 3. [Sch08] If λ^ is an algebraically simple eigenvalue of T (·), then λ^ is geometrically simple. 4. [Neu85, Sch08] Let (y^; λ,^ x^) be an eigentriplet of T (·). Then λ^ is algebraically simple if and only if λ^ is geometrically simple and y^∗T 0(λ^)x^ 6= 0. 5. [Sch08] Let D ⊂ C and E ⊂ Cd be open sets. Let T : D ×E ! Cn×n be continuously differentiable, and let λ^ be a simple eigenvalue of T (·; 0) and x^ and y^ right and left eigenvectors with unit norm. Then the first order perturbation expansion at λ^ reads as follows: d ^ 1 X ∗ @T ^ λ(") − λ = "jy^ (λ, 0)x^ + o(k"k): ∗ 0 ^ @" y^ T (λ, 0)x^ j=1 j Nonlinear Eigenvalue Problem 115-3 The normwise condition number for λ^ is given by v u d 2 jλ(") − λ^j 1 uX @T κ(λ^) = lim sup = t y^∗ (λ,^ 0)x^ : k"k ∗ 0 ^ @" k"k!0 jy^ T (λ, 0)x^j j=1 j 6. [Sch08] Let (y^; λ,^ x^) be an eigentriplet of T (·) with simple eigenvalue λ^. Then for sufficiently small jλ^ − λj 1 x^y^∗ T (λ)−1 = + O(1): λ − λ^ y^∗T 0(λ^)x^ 7. [Neu85] Let λ^ be a simple eigenvalue of T (·), and let x^ be a right eigenvector normal- ized such that e∗x^ = 1 for some vector e. Then the matrix B := T (λ^) + T 0(λ^)xe^ ∗ is nonsingular. 8. If T (·) is real symmetric and λ is a real eigenvalue, then left and right eigenvectors corresponding to λ coincide. 9. If T (·) is complex symmetric and x is a right eigenvector, then x¯ is a left eigenvector corresponding to the same eigenvalue. 10. If T (·) is Hermitian, then eigenvalues are real (and left and right eigenvectors corre- sponding to λ coincide) or they come in pairs, i.e. if (y; λ, x) is an eigentriplet of T (·), then this is also true for (x; λ,¯ y). Examples: 1. For the quadratic eigenvalue problem T (λ)x = 0 with 0 1 7 −5 1 0 T (λ) := + λ + λ2 (115.2) −2 3 10 −8 0 1 the distinct eigenvalues λ = 1 and λ = 2 share the eigenvector [1; 2]. " 2 # eiλ 1 p 2. Let T (λ)x := x = 0. Then T (λ)x = 0 has a countable set of eigenvalues 2kπ, 1 1 ^ k 2 N [ f0g. λ = 0 is an algebraically double and geometrically simple eigenvaluep with ∗ 0 ^ left and right eigenvectors x^ = y^ = [1; −1], and y^ T (0)x^ = 0. Every λk = 2kπ, k 6= 0 is algebraicallyp and geometrically simple with the same eigenvectors x^; y^ as before, and ∗ 0 ^ y^ T (λk)x^ = 2 2kπi 6= 0. k 3. T (λ) = (λ ), k 2 N has the eigenvalue λ^ = 0 with algebraic multiplicity k. 115.2 Analytic matrix functions In this section we consider the eigenvalue problem (115.1) where T (·): D ! Cn×n is a regular matrix function which is analytic in a neighborhood of an eigenvalue λ^. Definitions: A sequence of vectors x0; x1;:::; xr−1 is called a Jordan chain (of length r) corresponding to λ^ if x0 6= 0 and ` k X 1 d T (λ^) x = 0 for ` = 0; : : : ; r − 1: k `−k k! dλ ^ k=0 λ=λ 115-4 Handbook of Linear Algebra x0 is an eigenvector and x1;:::; xr−1 are generalized eigenvectors. Let x0 be an eigenvector corresponding to an eigenvalue λ^. The maximal length of a Jordan chain that starts with x0 is called the multiplicity of x0. An eigenvalue λ^ is is said to be normal if it is a discrete point in σ(T (·))) and the multiplicity of each corresponding eigenvector is finite. n An analytic function x : D ! C is called root function of T (·) at λ^ 2 D if T (λ^)x(λ^) = 0 and x(λ^) 6= 0. The multiplicity of λ^ as a zero of T (λ)x(λ) is called the multiplicity of x(·). The rank of an eigenvector x0 is the maximum of the multiplicities of all root functions x(·) such that x(λ^) = x0. A root function x(·) is called a maximal root function if the multiplicity of x(·) is equal to the rank of x0 := x(λ^). (1) ^ (1) P1 (1) ^ j Let x0 2 ker T (λ) be an eigenvector with maximal rank and let x (λ) = j=0 xj (λ − λ) (1) ^ (1) (k) be a maximal root function such that x (λ) = x0 .