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1 Electronic Transactions on Numerical Analysis. Volume 40, pp , Copyright 2013,. ISSN ETNA ON SYLVESTER S LAW OF INERTIA FOR NONLINEAR EIGENVALUE PROBLEMS ALEKSANDRA KOSTIĆ AND HEINRICH VOSS Dedicated to Lothar Reichel on the occasion of his 60th birthday Abstract. For Hermitian matrices and generalized definite eigenproblems, the LDL H factorization provides an easy tool to slice the spectrum into two disjoint intervals. In this note we generalize this method to nonlinear eigenvalue problems allowing for a max characterization of (some of) their real eigenvalues. In particular we apply this approach to several classes of quadratic pencils. Key words. eigenvalue, variational characterization, max principle, Sylvester s law of inertia AMS subject classification. 15A18, 65F15 1. Introduction. The inertia of a Hermitian matrix A is the triplet of nonnegative integers In(A) := (n p,n n,n z ), where n p, n n, and n z are the number of positive, negative, and zero eigenvalues of A (counting multiplicities). Sylvester s classical law of inertia states that two Hermitian matricesa,b C n n are congruent (i.e.,a = S H BS for some nonsingular matrixs) if and only if they have the same inertia In(A) = In(B). An obvious consequence of the law of inertia is the following corollary: if A has an LDL H factorization A = LDL H, then n p and n n equal the number of positive and negative entries of D, and if a block LDL H factorization exists where D is a block diagonal matrix with 1 1 and indefinite 2 2 blocks on its diagonal, then one has to increase the number of positive and negative 1 1 blocks of D by the number of 2 2 blocks to get n p and n n, respectively. Hence, the inertia of A can be computed easily. This is particularly advantageous if the matrix is banded. If B C n n is positive definite, and A σb = LDL H is the block diagonal LDL H factorization of A σb for some σ R, we get the inertia In(A σb) = (n p,n n,n z ) as described in the previous paragraph. Then, the generalized eigenvalue problem Ax = λbx has n n eigenvalues smaller than σ. Hence, the law of inertia yields a tool to locate eigenvalues of Hermitian matrices or definite matrix pencils. Combining it with bisection or the secant method, one can detere all eigenvalues in a given interval or detere initial approximations for fast eigensolvers, and it can be used to check whether a method has found all eigenvalues in an interval of interest or not. The law of inertia was first proved in 1858 by J. J. Sylvester [19], and several different proofs can be found in textbooks [3, 6, 11, 13, 15], one of which is based on the max characterization of eigenvalues of Hermitian matrices. In this note we discuss generalizations of the law of inertia to nonlinear eigenvalue problems allowing for a max characterization of their eigenvalues. 2. Minmax characterization. Our main tools in this paper are variational characterizations of eigenvalues of nonlinear eigenvalue problems generalizing the well known max characterization of Poincaré [16] or Courant [2] and Fischer [5] for linear eigenvalue problems. Received October 17, Accepted January 29, Published online on April 22, Recommended by D. Kressner. Ma sinski fakultet Sarajevo, Univerzitet Sarajevo, Bosnia and Herzegovina, (kostic@mef.unsa.ba). Institute of Mathematics, Hamburg University of Technology, D Hamburg, Germany (voss@tu-harburg.de). 82

2 SYLVESTER S LAW OF INERTIA FOR NONLINEAR EIGENPROBLEMS 83 We consider the nonlinear eigenvalue problem (2.1) T(λ)x = 0, where T(λ) C n n, λ J, is a family of Hermitian matrices depending continuously on the parameter λ J, and J is a real open interval which may be unbounded. To generalize the variational characterization of eigenvalues, we need a generalization of the Rayleigh quotient. To this end we assume that (A 1 ) for every fixed x C n, x 0, the scalar real equation (2.2) f(λ;x) := x H T(λ)x = 0 has at most one solution λ =: p(x) J. Then the equation f(λ;x) = 0 implicitly defines a functional p on some subset D C n, which is called the Rayleigh functional of (2.1), and which is exactly the Rayleigh quotient in case of a monic linear matrix function T(λ) = λi A. Generalizing the definiteness requirement for linear pencilst(λ) = λb A, we further assume that (A 2 ) for every x D and every λ J withλ p(x) it holds that (λ p(x))f(λ;x) > 0. If p is defined on D = C n \ {0}, then the problem T(λ)x = 0 is called overdamped. This notion is motivated by the finite dimensional quadratic eigenvalue problem T(λ)x = λ 2 Mx+λCx+Kx = 0, where M, C, and K are Hermitian and positive definite matrices. If C is large enough such thatd(x) := () 2 4(x H Kx)(x H Mx) > 0 for everyx 0, thent( ) is overdamped. Generalizations of the max and max characterizations of eigenvalues were proved by Duffin [4] for the quadratic case and by Rogers [17] for general overdamped problems. For nonoverdamped eigenproblems, the natural ordering to call the smallest eigenvalue the first one, the second smallest the second one, etc., is not appropriate. This is obvious if we make a linear eigenvalue problemt(λ)x := (λi A)x = 0 nonlinear by restricting it to an interval J which does not contain the smallest eigenvalue of A. Then the conditions (A 1 ) and (A 2 ) are satisfied, p is the restriction of the Rayleigh quotient R A to D := {x 0 : R A (x) J}, and inf x D p(x) will in general not be an eigenvalue. If λ J is an eigenvalue of T( ), then µ = 0 is an eigenvalue of the linear problem T(λ)y = µy, and therefore there exists an l N such that 0 = max V H l v V\{0} v H T(λ)v v 2, where H l denotes the set of all l dimensional subspaces of C n. In this case, λ is called an lth eigenvalue of T( ). With this enumeration the following max characterization for eigenvalues was proved in [20, 21]. THEOREM 2.1. Let J be an open interval in R, and let T(λ) C n n, λ J, be a family of Hermitian matrices depending continuously on the parameter λ J such that the conditions (A 1 ) and (A 2 ) are satisfied. Then the following statements hold.

3 84 ALEKSANDRA KOSTIĆ AND HEINRICH VOSS (i) For every l N there is at most one lth eigenvalue of T( ) which can be characterized by (ii) If (2.3) λ l = sup p(v). V H l, V D v V D λ l := inf sup p(v) J V H l, V D v V D for somel N, then λ l is the lth eigenvalue oft( ) inj, and (2.3) holds. (iii) If there exist the kth and the lth eigenvalue λ k and λ l in J (k < l), then J contains thejth eigenvalue λ j (k j l) as well withλ k λ j λ l. (iv) Let λ 1 = inf x D p(x) J and λ l J. If the imum in (2.3) is attained for an l-dimensional subspace V, then V D {0}, and (2.3) can be replaced by λ l = sup V H l, V D {0} v V, v 0 p(v). (v) λ is an lth eigenvalue if and only if µ = 0 is the lth largest eigenvalue of the linear eigenproblem T( λ)x = µx. (vi) The imum in (2.3) is attained for the invariant subspace oft(λ l ) corresponding to itsllargest eigenvalues. 3. Sylvester s law for nonlinear eigenvalue problems. We first consider the overdamped case. Then T( ) has exactly n eigenvalues λ 1 λ n inj [17]. THEOREM 3.1. Assume that T : J C n n satisfies the conditions of the max characterization in Theorem 2.1, and assume that the nonlinear eigenvalue problem (2.1) is overdamped, i.e., for every x 0 Equation (2.2) has a unique solutionp(x) J. For σ J, let (n p,n n,n z ) be the inertia of T(σ). Then the nonlinear eigenproblem T(λ)x = 0 has n eigenvalues in J, n p of which are less than σ, n n exceed σ, and for n z > 0, σ is an eigenvalue of geometric multiplicityn z. Proof. The invariant subspace W of T(σ) corresponding to its positive eigenvalues has dimension n p, and it holds that f(σ;x) = x H T(σ)x > 0 for every x W, x 0. Hence p(x) < σ by (A 2 ), and therefore the n p th smallest eigenvalue of T( ) satisfies λ np = dimv=n p max p(x) max p(x) < σ. x V, x W, On the other hand for every subspace V of C n of dimension n p + n z + 1, there exists a vector x V such that f(σ; x) < 0. Thus p(x) > σ, and it holds that λ np+n z+1 = dimv=n p+n z+1 max p(x) > σ, x V, which completes the proof. Next we consider the case that an extreme eigenvalue, i.e., either λ 1 := inf x D p(x) or λ n := sup x D p(x), is contained inj. THEOREM 3.2. Assume that T : J C n n satisfies the conditions of the max characterization, and let (n p,n n,n z ) be the inertia of T(σ) for someσ J. (i) If λ 1 := inf x D p(x) J, then the nonlinear eigenproblem T(λ)x = 0 has exactly n p eigenvalues λ 1 λ np inj which are smaller than σ. (ii) If sup x D p(x) J, then the nonlinear eigenproblem T(λ)x = 0 has exactly n n eigenvalues λ n nn+1 λ n inj exceeding σ.

4 SYLVESTER S LAW OF INERTIA FOR NONLINEAR EIGENPROBLEMS 85 Proof. (i): We first show that f(λ;x) < 0 for every λ J with λ < λ 1 and for every vector x 0. Assume that f(λ;x) 0 for some λ < λ 1 and x 0, let ˆx be an eigenvector of (2.1) corresponding to λ 1, and let w(t) := tˆx + (1 t)x, 0 t 1. Thenφ(t) := f(λ;w(t)) is continuous in[0,1],φ(0) = f(λ;x) 0, andφ(1) = f(λ;ˆx) < 0 by (A 2 ). Hence, there exists a value ˆt [0,1) such that f(λ;w(ˆt)) = 0, i.e., w(ˆt) D and p(w(ˆt)) = λ < λ 1 contradicting λ 1 := inf x D p(x). For n p = 0, the matrix T(σ) is negative semidefinit, i.e., x H T(σ)x 0 for x 0, and it follows from(a 2 ) thatp(x) σ holds true for everyx D. Hence, there is no eigenvalue less than σ. For n p > 0, let W denote the invariant subspace of T(σ) corresponding to its positive eigenvalues. Then f(σ;x) = x H T(σ)x > 0 for x W, x 0, and from f(λ;x) < 0 forλ < λ 1, it follows thatx D andp(x) < σ. Hence,W D {0} and as in the proof of Theorem 3.1 we obtain λ np = dimv=n p, V D max p(x) max p(x) < σ, x V D x W, i.e.,t( ) has at least n p eigenvalues less than σ. Assume that there exists an (n p + 1)th eigenvalue λ np+1 < σ of T( ), and let W be the invariant subspace of T(λ np+1) corresponding to its nonnegative eigenvalues. Then this implies that dimw n p + 1, W \ {0} D, and p(x) λ np+1 < σ for every x W with x 0, contradicting the fact that for every subspace V with dimv = n p +1 there exists a vector x V withx H T(σ)x 0, i.e., either x D or p(x) σ. (ii)s(λ) := T( λ) satisfies the conditions of Theorem 2.1 in the interval J, and J contains the smallest eigenvalue λ n of S. For the general case the law of inertia has the following form: THEOREM 3.3. Let T : J C n n satisfy the conditions of the max characterization, and let σ,τ J,σ < τ. Let (n pσ,n nσ,n zσ ) and (n pτ,n nτ,n zτ ) be the inertias of T(σ) and T(τ), respectively. Then the inequalityn pσ n pτ holds, and the eigenvalue problem (2.1) has exactlyn pτ n pσ eigenvalues λ npσ +1 λ npτ in(σ,τ). Proof. Let W be the invariant subspace of T(σ) corresponding to its positive eigenvalues. Then by the positive definiteness, x H T(σ)x > 0, for every x W, x 0, it follows from (A 2 ) that x H T(τ)x > 0 holds as well, hence, n pσ n pτ. Let V be a subspace of C n with V D and dimv = n pσ +1. We first show that there exits ax V D withp(x) > σ, from which we then obtain λ npσ +1 := inf dimv=n pσ +1, V D sup p(x) > σ. x V D From the hypothesis dimv > n pσ, it follow that there exists a vector x V, x 0 such that x H T(σ)x < 0. If x D, then it follows from (A 2 ) that we are done. Otherwise, we choose y V D and ω > (p(y),σ). Then x H T(ω)x < 0 < y H T(ω)y, and with w(t) := tx + (1 t)y it follows in the same way as in the proof of Theorem 3.2 that there exist a value ˆt [0,1] such that w(ˆt) V D and p(w(ˆt)) = ω > σ. If U denotes the invariant subspace of T(τ) corresponding to its positive eigenvalues, then x H T(τ)x > 0 holds for every x U,x 0. If U D =, then x H T(λ)x > 0 holds for every λ J and x U, x 0 and in particular for λ = σ. Hence, U W and from σ < τ the equality n pσ = n pτ follows. In this case, T( ) has no eigenvalue in (σ,τ) because otherwise a corresponding eigenvector x would satisfyx H T(σ)x < 0 < x H T(τ)x, i.e., x W and x U contradicting U W.

5 86 ALEKSANDRA KOSTIĆ AND HEINRICH VOSS IfU D, then p(x) < τ holds for every x U D and therefore λ npτ = inf dimv=n pτ, V D sup p(x) sup p(x) < τ. x V D x U D Hence, λ npσ +1 and λ npτ are both contained in (σ,τ) and so are the eigenvalues λ j forj = n pσ +1,...,n pτ. REMARK 3.4. Without using the max characterization of eigenvalues, Neumaier [13] proved Theorem 3.3 for matrices T : J C n n which are Hermitian and (elementwise) differentiable in J with positive definite derivative T (λ), λ J. Obviously, such T( ) satisfy the conditions of the max characterization. EXAMPLE 3.5. Consider the rational eigenvalue problem T(λ) := K +λm + p j=1 λ σ j λ C jc T j, where K,M R n n are symmetric and positive definite, the matrix C j R n kj has rank k j, and 0 < σ 1 < < σ p. This problem models the free vibrations of certain fluid solid structures; cf. [1]. In each interval J l := (σ l,σ l+1 ), l = 0,...,p, σ 0 = 0, σ p+1 =, the function f l (λ,x) := x H T(λ)x is strictly monotonically increasing, and therefore all eigenvalues inj l are max values of the Rayleigh functional p l. For the first interval J 0, Theorem 3.2 applies. Hence, if τ J 0 and (n p,n n,n z ) is the inertia of T(τ), then there are exactly n p eigenvalues in J 0 which are less than τ. Moreover, if τ 1 < τ 2 are contained in one interval J j, then the number of eigenvalues in the interval (τ 1,τ 2 ) can be obtained from the inertias of T(τ 1 ) and T(τ 2 ) according to Theorem Quadratic eigenvalue problems. We consider quadratic matrix pencils (4.1) Q(λ) := λ 2 A+λB +C, with Hermitian matricesa,b,c C n n under several conditions that guarantee that (some of) the real eigenvalues allow for a variational characterization and hence for slicing of the spectrum using the inertia C < 0 and A 0. Let C be negative definite and A positive semidefinite. Multiplying Q(λ)x = 0 byλ 1, one gets the equivalent nonlinear eigenvalue problem Q(λ)x := λax+bx+λ 1 Cx = 0. Differentiating f(λ;x) := x H Q(λ)x with respect toλyields λ f(λ;x) = xh Ax λ 2 > 0 for every x 0 and every λ 0. Hence, Q satisfies the conditions of the max characterization for both intervals J := (,0) and J + := (0, ). For the corresponding Rayleigh functional p ± with domain D ±, it holds that λ + 1 = inf x D + p + (x) J + and λ n = sup x D p (x) J, and therefore the following statement follows from Theorem 3.2. THEOREM 4.1. Let C be negative definite and A positive semidefinite. (i) For σ > 0 let In( Q(σ)) = (n p,n n,n z ) be the inertia of Q(σ). Then the quadratic pencil (4.1) has n p positive eigenvalues less than σ.

6 SYLVESTER S LAW OF INERTIA FOR NONLINEAR EIGENPROBLEMS 87 (ii) For σ < 0 let In( Q(σ)) = (n p,n n,n z ) be the inertia of Q(σ). Then (4.1) has n n negative eigenvalues exceeding σ. If A is positive definite, then Q is overdamped with respect to J + and J, and therefore there exist exactly n positive and n negative eigenvalues. If A 0 is positive semidefinite andr = rank(a), then is an infinite eigenvalue of multiplicityn r, and there are onlyn+r finite eigenvalues. IfB is positive definite, then the Rayleigh functional p + (x) = 2 x H Bx+ (x H Bx) 2 4(x H Ax)() is defined on C n \ {0}. Hence, ( Q,J + ) is overdamped, and there exist n positive and r negative eigenvalues. Theorem 4.1 can be strengthened according to the following result. THEOREM 4.2. Assume that A is positive semidefinite, B is positive definite, and C is negative definite. (i) For σ > 0 let In( Q(σ)) = (n p,n n,n z ) be the inertia of Q(σ). Then the quadratic pencil (4.1) has n p positive eigenvalues less than σ, n n eigenvalues exceeding σ, and ifn z 0, thenσ is an eigenvalue of Q( ) with multiplicityn z. (ii) For σ < 0 let In( Q(σ)) = (n p,n n,n z ) be the inertia of Q(σ). Then (4.1) has n n negative eigenvalues exceedingσ,n p r finite eigenvalues less thanσ, and ifn z 0, then σ is an eigenvalue of Q( ) with multiplicityn z Hyperbolic problems. The quadratic pencil Q( ) defined by the Hermitian matrices A,B,C C n n is called hyperbolic if A is positive definite and for every x C n, x 0, the quadratic polynomial has two distinct real roots (4.2) p ± (x) = xh Bx 2x H Ax ± f(λ;x) := λ 2 x H Ax+λx H Bx+ = 0 ( ) 2 xh Bx 2x H xh Cx Ax x H Ax. A hyperbolic quadratic matrix polynomial Q( ) has the following properties (cf. [12]): the ranges J ± := p ± (C n \ {0}) are disjoint real closed intervals with max J < J +, moreover Q(λ) is positive definite for λ < J and λ > max J +, and it is negative definite forλ (max J, J + ). Let J + be an open interval with J + J + and J J + =, and let J be an open interval with J J and J + J =. Then(Q,J + ) and( Q,J ) satisfy the conditions of the variational characterization of eigenvalues and they are both overdamped. Hence, there exist 2n eigenvalues and λ 1 λ n < λ n+1 λ 2n λ j = dimv=j max p (x) and x V, λ n+j = dimv=j max p +(x), j = 1,...,n. x V, If In(Q(σ)) = (n p,n n,n z ) is the inertia of Q(σ) and n n = n, then Q(σ) is negative definite and there are n eigenvalues smaller than σ and n eigenvalues exceeding σ. If n p = n holds, then Q(σ) is positive definite, f(σ;x) > 0 is valid for every x 0, and if λ f(σ;x) < 0 holds, then it follows that σ < λ 1, and σ > λ 2n otherwise.

7 88 ALEKSANDRA KOSTIĆ AND HEINRICH VOSS If n p n and n n n, then σ J J + and Theorem 3.1 applies. We only have to find out which of these intervals σ is located in. To this end we detere x 0 such thatf(σ;x) := x H Q(σ)x > 0 (this can be done by a few steps of the Lanczos method which is known to converge first to extreme eigenvalues). If λ f(σ;x) = 2σxH Ax+x H Bx < 0, then it follows that p (x) > σ, and therefore σ < λ n = max p (x). Similarly, the inequalites f(σ;x) > 0 and 2σx H Ax + x H Bx > 0 imply σ > λ n+1 = p + (x). Hence we obtain the following slicing of the spectrum of Q( ). THEOREM 4.3. Let Q(λ) := λ 2 A+λB +C be hyperbolic, and let(n p,n n,n z ) be the inertia of Q(σ) forσ R. (i) If n n = n then there are n eigenvalues smaller than σ and n eigenvalues greater than σ. (ii) Let n p = n. If 2σx H Ax + x H Bx < 0 for an arbitrary x 0, then there are 2n eigenvalues exceeding σ. If 2σx H Ax+x H Bx > 0 for an arbitrary x 0, then all 2n eigenvalues are less than σ. (iii) For n p = 0 and n z > 0 let x 0 be an element of the null space of Q(σ). If 2σx H Ax+x H Bx < 0, then Q(λ)x = 0 has n n z eigenvalues in (,σ), n eigenvalues in(σ, ) and σ = λ n with multiplicityn z. If 2σx H Ax+x H Bx > 0, then Q( ) has n eigenvalues in (,σ), n n z eigenvalues in(σ, ), and σ = λ n+1 with multiplicityn z. (iv) For n p > 0 and n z = 0 let x 0 be such that f(σ;x) > 0. If2σx H Ax+x H Bx < 0, thenq( ) hasn n p eigenvalues in(,σ) andn+n p eigenvalues in(σ, ). If2σx H Ax+x H Bx > 0, thenq( ) hasn+n p eigenvalues in(,σ) andn n p eigenvalues in(σ, ). (v) For n p > 0 and n z > 0 let x 0 be such that f(σ;x) > 0. If 2σx H Ax + x H Bx < 0, then Q( ) has n n p n z eigenvalues in (,σ) and n+n p eigenvalues in(σ, ). If 2σx H Ax + x H Bx > 0, then Q( ) has n + n p eigenvalues in (,σ) and n n p n z eigenvalues in(σ, ). In either case σ is an eigenvalue with multiplicityn z. REMARK 4.4. These results on quadratic hyperbolic pencils can be generalized to a hyperbolic matrix polynomial of higher degree P(λ) = k λ j A j, A j = A H j, j = 0,...,k, A k > 0, j=0 which is hyperbolic if A k is positive definite and for every x 0 the corresponding polynomial f(λ;x) := x H P(λ)x has k real and distinct roots. In this case there exist k disjoint open intervals J j R, j = 1,...,k such that P( ) has exactlyneigenvalues in eachj j, and these eigenvalues allow for a max characterization; cf. [12, 14]. To fix the numeration let supj j+1 < infj j for j = 1,...,k 1. For σ R, let (n p,n n,n z ) be the inertia of P(σ), and let x C n be a vector such that x H P(σ)x > 0. Iff( ;x) has exactly j roots which exceed σ, then it holds that σ J j+1 or σ [supj j+1,infj j ] or σ J j.

8 SYLVESTER S LAW OF INERTIA FOR NONLINEAR EIGENPROBLEMS 89 Which one of these situations occur can be deduced from the inertia (n n = n orn p = n) and the derivative λf(σ;x) as for the quadratic case Definite quadratic pencils. In a recent paper, Higham, Mackey, and Tisseur [10] generalized the concept of hyperbolic quadratic polynomials waiving the positive definiteness of the leading matrixa. A quadratic pencil (4.1) is definite if A, B, and C are Hermitian, there exists a real value µ R { } such that Q(µ) is positive definite, and for every x C n, x 0 the scalar quadratic polynomial f(λ;x) := λ 2 x H Ax+λx H Bx+ = 0 has two distinct roots inr { }. The following theorem was proved in [10]. THEOREM 4.5. The Hermitian matrix polynomialq(λ) is definite if and only if any two (and hence all) of the following properties hold: d(x) := (x H Bx) 2 4(x H Ax)() > 0 for every x C n \{0}, Q(η) > 0 for some η R { }, Q(ξ) < 0 for some ξ R { }. For ξ < η (otherwise consider Q( λ)) it was shown in [14] that there are n eigenvalues in(ξ,η) which are max values of a Rayleigh functional, and the remainingneigenvalues in [,ξ) and (η, ] are max and max values of a second Rayleigh functional. Hence, if ξ and η are known, then the slicing of the spectrum using the LDL H factorization follows similarly to the hyperbolic case. However, given σ and the LDL H factorization ofq(σ), we are not aware of an easy way to decide in which of the intervals[,ξ),(ξ,η), or(η, ] the parameterσ is located. The articles [7, 8, 14] contain methods to detect whether a quadratic pencil is definite and to compute the parameters ξ and η, however they are much more costly than computing anldl H factorization of a matrix. For the Examples 4.6 and 4.8 at least one of these parameters are known and the slicing can be given explicitly. EXAMPLE 4.6. Duffin [4] called a quadratic eigenproblem (4.1) an overdamped network, ifa, B, and C are positive semidefinite and the so called overdamping condition d(x) = (x H Bx) 2 4(x H Ax)() > 0 for every x 0 is satisfied. So, actually B has to be positive definite, and therefore Q(µ) is positive definite for every µ > 0, and Q( ) is definite. Ifr denotes the rank ofa, then (4.1) hasn+r finite real eigenvalues, the largestnones of which (called primary eigenvalues by Duffin) are max values of p + (x) := 2 x H Bx+ d(x), and the smallestr ones (called secondary eigenvalues) are max values λ n+1 j = max p (x), dimv=j, V D x V D ( where D := {x C n : x H Ax 0} and p (x) = x H Bx ) d(x) /(2x H Ax) for x D. Hence, the following slicing of the spectrum can be derived. THEOREM 4.7. LetA,B,C C n n be positive semidefinite and assume thatd(x) > 0 for x 0. Let r be the rank of A and In(Q(σ)) = (n p,n n,n z ) be the inertia of Q(σ) for some σ R. Then the following holds.

9 90 ALEKSANDRA KOSTIĆ AND HEINRICH VOSS (i) If n n = n, then there are r eigenvalues smaller than σ and n eigenvalues greater than σ. (ii) For n p = 0 and n z > 0 let x 0 be an element of the null space of Q(σ). If 2σx H Ax + x H Bx < 0, then Q( ) has r n z eigenvalues in (,σ) and n eigenvalues in(σ,0]. If 2σx H Ax + x H Bx > 0, then Q( ) has r eigenvalues in (,σ), and n n z eigenvalues in(σ,0]. In either case, σ is an eigenvalue of Q( ) with multiplicityn z. (iii) For n p > 0 let x 0 be such that f(σ;x) > 0. If 2σx H Ax + x H Bx < 0, then Q(λ)x = 0 has r n p eigenvalues in (,σ) and n+n p n z eigenvalues in(σ,0]. If2σx H Ax+x H Bx > 0, thenq(λ)x = 0 hasr+n p n z eigenvalues in(,σ) and n n p eigenvalues in(σ,0]. EXAMPLE 4.8. Free vibrations of fluid solid structures are governed by the nonsymmetric eigenvalue problem [9, 18] [ ][ ] [ ][ ] Ks C xs Ms 0 xs (4.3) = λ 0 K f x f C T x f where K s R s s, K f R f f are the stiffness matrices, and M s R s s, M f R f f are the mass matrices of the structure and the fluid, respectively, andc R s f describes the coupling of structure and fluid. The vector x s is the structure displacement vector, and x f is the fluid pressure vector. K s, M s,k f, and M f are symmetric and positive definite. Multiplying the first line of (4.3) by λ, one obtains the quadratic pencil [ ] [ ] [ ] Q(λ) := λ 2 Ms 0 Ks C 0 0 +λ 0 0 C T +. M f 0 K f [ ] It can easily be seen that for x s 0 the quadratic equation [x T s,x T f ]Q(λ) xs = 0 has one positive solution p + (x s,x f ) and one negative solution p (x s,x f ), and for x s = 0 it has one positive solutionp + (x s,x f ) := x T f K fx f /(x T f M fx f ) and the solutionp (x s,x f ) :=. Moreover the positive eigenvalues of (4.3) are max values of the Rayleigh functionalp +. Hence, one gets the following result for the (physically meaningful) positive eigenvalues: if In(Q(σ)) = (n p,n n,n z ) for σ > 0, then there are exactly n p eigenvalues in (0,σ), n n eigenvalues in(σ, ), and ifn z 0, then σ is an eigenvalue of multiplicityn z Nonoverdamped quadratic pencils. We consider the quadratic pencil (4.1) where the matrices A, B, and C are positive definite. Then for x 0, the two complex roots of f(λ;x) := x H Q(λ)x are given in (4.2). Let us define δ := sup{p (x) : p (x) R}, M f x f δ + := inf{p + (x) : p + (x) R}, J := (,δ + ), J + = (δ,0), and D ± := {x C n : p ± (x) J ± }. If f(λ,x) > 0 for x 0 and λ R, then it follows that δ = and δ + =. The eigenvalue problem Q(λ)x = 0 has no real eigenvalues, but this does not have to be known in advance. Theorem 4.9 applies to this case as well. It is obvious that Q and Q satisfy the conditions of the max characterization of its eigenvalues in J and J +, respectively. Hence, all eigenvalues in J are max-values of p : λ j = dimv=j, V D max p (x), j = 1,2,... x V D

10 SYLVESTER S LAW OF INERTIA FOR NONLINEAR EIGENPROBLEMS 91 Taking advantage of the max characterization of the eigenvalues of Q(λ) := Q( λ) in J := J + with the Rayleigh functional p := p +, we obtain the following max characterization λ + 2n+1 j = max dimv=j, V D + p + (x), j = 1,2,... x V D + of all eigenvalues of Q inj +. Hence, for σ < δ + and for σ > δ, we obtain slicing results for the spectrum of Q( ) from Theorem 3.2. If In(Q(σ)) = (n p,n n,n z ) and σ < δ +, then there exist n n eigenvalues ofq( ) in(,σ), and ifσ (δ,0), then there aren n eigenvalues in(σ,0). However, δ + and δ are usually not known. The following theorem contains upper bounds of δ and lower bounds of δ +, thus yielding subintervals of (,δ + ) and (δ,0) where the above slicing applies. THEOREM 4.9. Let A,B,C C n n be positive definite, and let p + and p be defined in (4.2). Then it holds that (i) δ + := max x H Ax δ + = inf{p + (x) : p + (x) R} and δ = sup{p (x) : p (x) R} x H Ax =: δ. (ii) ˆδ + := 2max x H Bx δ + and δ 2 x H Bx =: ˆδ. Proof: (i): The value δ + is a lower bound of δ + if for every x 0 such that p + (x) R it holds that p + (x) δ +. The following proof takes advantage of the facts that p + (x) < 0 and λ f(p +(x);x) 0. The equation f(p + (x);x) = x H Q(p + (x))x = 0 is satisfied if and only if Hence, x H Bx = p + (x)x H Ax 1 p + (x) xh Cx. λ f(p +(x);x) = 2p + (x)x H Ax+x H Bx = p + (x)x H Ax 1 p + (x) xh Cx 0 if and only if p + (x) 2 xh Cx x H Ax, i.e., δ + max x H Ax = δ +, and analogously we obtain δ x H Ax = δ.

11 92 ALEKSANDRA KOSTIĆ AND HEINRICH VOSS (ii): Solving f(p + (x);x) = 0 for x H Ax, one gets from λ f(p +(x);x) 0 that and analogously p + (x) 2 xh Cx x H Bx, i.e.,δ + 2max x H Bx = ˆδ +, δ 2 x H Bx = ˆδ. From Theorem 3.2 we obtain the following slicing of the spectrum of Q( ). THEOREM Let A, B, and C be positive definite matrices, and for some σ R let In(Q(σ)) = (n p,n n,n z ). (i) If σ max { max x H Ax, 2max } x H, Bx then there exist n n eigenvalues of Q(λ)x = 0 in(,σ). (ii) If { } x σ H Cx x H Ax, 2 x H, Bx then there exist n n eigenvalues of Q(λ)x = 0 in(σ,0). EXAMPLE In a numerical experiment, the matrices A, B, and C were generated by the following MATLAB statements: randn( state,0); A=eye(20); B=randn(20);B=B *B; C=randn(20);C=C *C;. It was found that Q(λ)x = 0 has 26 real eigenvalues, 13 in the domain of p and 13 in the domain of p +. So, Sylvester s theorem can be applied to all of them. 12 eigenvalues are less than max(λ(c,a)) and 6 eigenvalues exceed (λ(c,a)). Conclusions. We have considered a given family of Hermitian matrices T(λ) depending continuously on a parameter λ in an open real interval J which allows for a variational characterization of its eigenvalues. We proved slicing results for the spectrum oft( ), where at first general nonlinear eigenvalue problems are considered, which are then specialized to various types of quadratic eigenproblems. REFERENCES [1] C. CONCA, J. PLANCHARD, AND M. VANNINATHAN, Fluid and Periodic Structures, vol. 38 of Research in Applied Mathematics, Masson, Paris, [2] R. COURANT, Über die Eigenwerte bei den Differentialgleichungen der mathematischen Physik, Math. Z., 7 (1920), pp [3] J. DEMMEL, Applied Numerical Linear Algebra, SIAM, Philadelphia, [4] R. DUFFIN, A imax theory for overdamped networks, J. Rational Mech. Anal., 4 (1955), pp [5] E. FISCHER, Über quadratische Formen mit reellen Koeffizienten, Monatsh. Math. Phys., 16 (1905), pp [6] G. GOLUB AND C. VAN LOAN, Matrix Computations, 3rd ed., John Hopkins University Press, Baltimore, [7] C.-H. GUO, N. HIGHAM, AND F. TISSEUR, Detecting and solving hyperbolic quadratic eigenvalue problems, SIAM J. Matrix Anal. Appl., 30 (2009), pp

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