The Hyperbolic Quadratic Eigenvalue Problem

Transcription

1 The Hyperboic Quadratic Eigenvaue Probem Xin Liang Ren-Cang Li Technica Report

2 The Hyperboic Quadratic Eigenvaue Probem Xin Liang Ren-Cang Li January 8, 2014 Abstract The hyperboic quadratic eigenvaue probem (HQEP) was shown to admit the Courant-Fischer type min-max principes in 1955 by Duffin and Cauchy type interacing inequaities in 2010 by Veseić. It can be regarded as the cosest anaogue (among a kinds of quadratic eigenvaue probems) to the standard Hermitian eigenvaue probem (among a kinds of standard eigenvaue probems). In this paper, we conduct a systematic study on HQEP both theoreticay and numericay. In the theoretic front, we generaize Wiedandt-Lidskii type min-max principes and, as a specia case, Ky-Fan type trace min/max principes and estabish Wey type and Mirsky type perturbation resuts when an HQEP is perturbed to another HQEP. In the numerica front, we justify the natura generaization of the Rayeigh-Ritz procedure with the existing and our new optimization principes and, as consequences of these principes, we extend various current optimization approaches steepest descent/ascent and noninear conjugate gradient type methods for the Hermitian eigenvaue probem to cacuate few extreme quadratic eigenvaues (of both pos- and neg-type). A detaied convergent anaysis is given on the steepest descent/ascent methods. The anaysis reveas the intrinsic quantities that contro convergence rates and consequenty yieds ways of constructing effective preconditioners. Numerica exampes are presented to demonstrate the proposed theory and agorithms. Key words. Hyperboic quadratic eigenvaue probem, Rayeigh quotient, min-max principe, Cauchy interacing inequaity, eigenvaue perturbation, extended steepest descent/ascent method, ocay optima extended conjugate gradient method, preconditioning AMS subject cassifications. 15A18, 15A42, 65F08, 65F30, 65G99 Schoo of Mathematica Sciences, Peking University, Beijing, , P. R. China. E-mai: iangxinsm@pku.edu.cn. Supported in part by China Schoarship Counci and Nationa Natura Science Foundation of China NSFC This work is primariy done whie this author was a visiting student, from August 2011 to September 2013, at Department of Mathematics, University of Texas at Arington, Arington, TX Department of Mathematics, University of Texas at Arington, P.O. Box 19408, Arington, TX E-mai: rci@uta.edu. Supported in part by NSF grants DMS and DMS , and a Research Gift Grant from Inte Corporation. 1

3 Contents 1 Introduction 3 2 Hyperboic quadratic matrix poynomia 5 3 Variationa principes Courant-Fischer type min-max principes Wieandt-Lidskii type min-max principes Cauchy type interacing inequaities Perturbation anaysis Setting the stage Asymptotica anaysis Perturbation bounds in the spectra norm Perturbation bounds in unitariy invariant norms Perturbation bounds in the Frobenius norms Best approximations from a subspace and Rayeigh-Ritz procedure 42 6 The steepest descent/ascent method Gradients The steepest descent/ascent method The extended steepest descent/ascent method Convergence anaysis Proof of Theorem Preconditioned steepest descent/ascent method Preconditioning Preconditioned steepest descent/ascent method Convergence anaysis Bock preconditioned steepest descent/ascent method 71 9 Conjugate gradient method Preconditioned conjugate gradient method Locay optima bock preconditioned extended conjugate gradient method Numerica exampes Concuding remarks 81 A Digression: positive semidefinite matrix penci 83 2

4 1 Introduction It was argued in [26] that the hyperboic quadratic eigenvaue probem (HQEP) is the cosest anaogue of the standard Hermitian eigenvaue probem when it comes to the quadratic eigenvaue probem (QEP) (λ 2 A + λb + C)x = 0. (1.1) In many ways, both probems share common properties: the eigenvaues are a rea, and for HQEP there is a version of the min-max principes [12, 1955] that is very much ike the Courant-Fischer min-max principes. One source of QEPs (1.1) is dynamica systems with friction, where A, C are associated with the kinetic-energy and potentia-energy quadratic form, respectivey, and B is associated with the Rayeigh dissipation function [16, 65]. When A, B, and C are Hermitian, and A and B are positive definite and C positive semidefinite, we say the dynamica system is overdamped if (x H Bx) 2 4(x H Ax)(x H Cx) > 0 for any nonzero vector x. Overdamped dynamica systems are common in eevator and car braking systems 1. A HQEP is sighty more genera than an overdamped QEP in that B and C are no onger required positive definite or positive semidefinite, respectivey. However, a a suitabe shift in λ can turn a HQEP into an overdamped QEP [20]. If (1.1) is satisfied for a scaar λ and nonzero vector x, we ca λ a quadratic eigenvaue, x an associated quadratic eigenvector, and (λ, x) a quadratic eigenpair. In this paper, we wi aunch a systematic study of the HQEP both in theory and numerica computations that wi further reinforce the beief that this cass of QEP is the cosest anaogue to the standard Hermitian eigenvaue probem. In the theoretica front, we wi review existing resuts of Courant-Fischer type min-max principes, Cauchy interacing inequaities; estabish Wieandt-Lidskii type min-max principes for the sums of seected quadratic eigenvaues and, as coroaries, trace min/max type principes; estabish perturbation resuts in the spectra and Frobenius norm, as we as genera unitariy invariant norms on how the quadratic eigenvaues wi change if A, B, C are perturbed. In the numerica front, we wi justify a naturay extended Rayeigh-Ritz type procedure, with the existing and newy estabished min-max principes, why the procedure wi produce the best approximations to quadratic eigenvaues/eigenvectors; propose extended steepest descent/ascent and CG type methods for computing extreme quadratic eigenpairs; 1 W. Kahan, private cmmunications, November

5 estabish convergence resuts, incuding the rate of convergence for the extended steepest descent/ascent methods, which shed ight on preconditioning in what constitutes a good preconditioner and how to construct one. In a separate paper, we wi extend most of the deveopment in this paper to the hyperboic poynomia eigenvaue probem. The rest of this paper is organized as foows. In section 2, we coect some properties for hyperboic quadratic matrix poynomias and estabish a few more about an HQEP. Wieandt-Lidskii type min-max principes, among others, are given in section 3. Eigenperturbation anaysis for HQEP is done in section 4. In section 5, we justify the use of the Rayeigh-Ritz procedure for extracting interested quadratic eigenvaues and their associated quadratic eigenvectors within a given subspace. The steepest descent/ascent method and its extended variation are studied in section 6, where a detaied convergence anaysis is performed. Section 7 investigates the preconditioning techniques to speed up the extended steepest descent/ascent method and expain how an effective preconditioner shoud be constructed from two different perspectives. Section 8 introduces the bock variations of the methods in the previous two sections. Various conjugate gradient methods the pain, ocay optima, and extended subspace search versions combined with suitabe preconditoners and bocking are described in detai in section 9. Two numerica exampes are presented in section 10 to demonstrate the effectiveness of the ocay optima bock preconditioned conjugate gradient method in the previous section. Finay in section 11, we present our concuding remarks. In appendix section A, we review the Jordan canonica form of a positive semidefinite matrix penci and estabish a perturbation theory for a positive definite matrix penci for use in section 4. Notation. Throughout this paper, C n m is the set of a n m compex matrices, C n = C n 1, and C = C 1. R is the set of a rea numbers. I n (or simpy I if its dimension is cear from the context) is the n n identity matrix, and e j is its jth coumn. X H is the conjugate transpose of a vector or matrix. For X C n m, σ min (X) is the smaest singuar vaue of X (X has min{m, n} singuar vaues), X 2 and X F and X ui are the spectra, Frobenius, and a genera unitariy invariant norm of X, and κ 2 (X) = X 2 X 1 2 is the condition number of X. A 0 (A 0) means that A is Hermitian positive (semi-)definite, and A 0 (A 0) if A 0 ( A 0). A 1/2 0 is the unique square root of A 0. The integer tripet (i (H), i 0 (H), i + (H)) denotes the inertia of an Hermitian matrix H, meaning that H has i (H) negative, i 0 (H) zero, and i + (H) positive eigenvaues, respectivey, and λ min (H) and λ max (H) are its smaest and argest eigenvaue. Generic notation eig( ) is the set of a eigenvaues, counting agebraic mutipicities, of a matrix or a matrix penci, depending on its argument(s): eig(a) is for A, and eig(a, B) is for A λb. We use poyeig(a 0, A 1,, A k ) as MATLAB s function poyeig for the set of a poynomia eigenvaues of λ k A k + + λa 1 + A 0. Note poyeig(a 0, A 1 ) is not the same of eig(a 0, A 1 ). 4

6 2 Hyperboic quadratic matrix poynomia Given A, B, C C n n, define a quadratic matrix poynomia of order n. Q(λ) := λ 2 A + λb + C, (2.1) Definition 2.1. Q(λ) is said Hermitian if A, B, and C are a Hermitian, hyperboic if it is Hermitian, A 0, and (x H Bx) 2 4(x H Ax)(x H Cx) > 0, for a 0 x C n, (2.2) overdamped if it is hyperboic as we as B 0, C 0. For a hyperboic Q(λ), define ς(x) := [ (x H Bx) 2 4(x H Ax)(x H Cx) ] 1/2, ς0 (x) := ς(x) x H x. (2.3) The quadratic eigenvaue probem (QEP) for Q( ) is to find λ C and 0 x C n such that Q(λ)x = 0. When this equation is satisfied, λ is caed a quadratic eigenvaue and x the associated quadratic eigenvector. Evidenty a quadratic eigenvaues of Q( ) is the roots of det Q(λ) = 0 which has 2n (compex) roots, counting mutipicities. The next theorem summarizes some of the reevant theoretica resuts on hyperboic quadratic poynomias. They can be found in Guo and Lancaster [20] which is an exceent gateway to references of origins for these resuts. Item 3(c) can be found in [64, (0.7)]. Theorem 2.1. Let Q(λ) = λ 2 A + λb + C as in (2.1) be Hermitian with A Q(λ) is hyperboic if and ony if there exists λ 0 R such that Q(λ 0 ) If Q(λ) is hyperboic, then its quadratic eigenvaues are a rea. 3. Suppose Q(λ) is hyperboic. Denote its quadratic eigenvaues by λ ± i and arrange them in the order of λ 1 λ n < λ + 1 λ+ n. (2.4) Then (a) Q(λ) 0 for a λ (λ n, λ + 1 ); (b) Q(λ) 0 for a λ (, λ 1 ) (λ+ n, + ); (c) the inertia of Q(λ) is (n k, 0, k) for λ (λ + k, λ+ k+1 ) or λ (λ n k, λ n+1 k ) for k = 1,, n, concuding that Q(λ) is indefinite for λ (λ typ 1, λ typ n ); (d) Q(λ) is overdamped if and ony if λ + n 0. 5

7 An immediate consequence of Theorem 2.1 is a test to determine whether Q(λ) is hyperboic or not [20]: check if its quadratic eigenvaues are a rea and, in the case they are a rea, check if Q(λ 0 ) 0, where λ 0 = (λ n + λ + 1 )/2. A common technique of soving QEP (1.1), or more generay the poynomia eigenvaue probem, is inearization that converts a poynomia eigenvaue probem to an equivaent generaized (inear) eigenvaue probem of a matrix penci [16, 25, 42]. Under the condition that A is nonsinguar, QEP (1.1) is equivaent to the generaized eigenvaue probem of the foowing matrix penci or L Q (λ) := [ ] C 0 0 A [ 0 C K Q (λ) := C B λ ] λ [ ] B A A 0 = A λb, (2.5) [ ] C 0 = A λb (2.6) 0 A in the sense that poyeig(c, B, A) = eig(a, B) and associated eigenvectors of one can be recovered from those for the other. More can be said if Q(λ) = λ 2 A+λB +C is hyperboic. Reevant resuts are summarized in the foowing emma, where item 5 is essentiay in [4] (see aso [9], [26, Theorem 3.6], and [63, Theorem 5A]). Theorem 2.2. Let Q(λ) = λ 2 A + λb + C as in (2.1) and et L Q (λ) be as in (2.5). Suppose A is nonsinguar. 1. poyeig(c, B, A) = eig(a, B). 2. If A 0 and B is Hermitian, then the inertia of B is (n, 0, n). [ ] x 3. If (µ, x) is an eigenpair of Q(λ), then (µ, ) is an eigenpair of L µx Q (λ). [ ] x 4. If (µ, ) is an eigenpair of L y Q (λ), then (µ, x) is an eigenpair of Q(λ) and y = µx. 5. Suppose Q(λ) is Hermitian. Q(λ) is hyperboic if and ony if L Q (λ) is a positive definite penci. 6. Suppose Q(λ) is hyperboic, and adopt the notation in item 3 of Theorem 2.1. Then L Q (λ) 0 for a λ (λ n, λ + 1 ). Proof. Since for any λ C, [ ] T [ ] [ ] [ ] I 0 Q(λ) 0 I 0 C λb λa = = L λi I 0 A λi I λa A Q (λ). (2.7) Thus ( 1) n det Q(λ) det A det L Q (λ) and item 1 foows. For item 2, A 0 guarantees that there is a nonsinguar matrix X C n n such that X H AX = I n, X H BX = diag(ω 1,..., ω n ) =: Ω, 6

8 where ω i R. We have [ X X ] H [ X B ] = X [ Ω ] In I n 0 (2.8) whose eigenvaues are the union of a the eigenvaues of [ ] ωi 1 for i = 1, 2,..., n. 1 0 But the two eigenvaues of each one of these 2 2 matrices are ω i ωi ω i + ωi 2 < 0, + 4 > Therefore the ast matrix in (2.8) has n positive and n negative eigenvaues, as expected. Items 3 and 4 can be verified in a straightforward way by using (2.7). Aso by using (2.7), we see that diag( Q(λ), A) and L Q (λ) are congruent for a λ R, and hence items 5 and 6 foow from items 1 and 3(a) of Theorem 2.1, respectivey. One consequence of Theorem 2.2 is that any hyperboic Q(λ) = λ 2 A + λb + C gives rise to a positive definite matrix penci L Q (λ) as defined by (2.5) with B having inertia (n, 0, n). There is a converse to the statement, too. Theorem 2.3. Let L(λ) = A λb be a positive definite Hermitian pair of order 2n. If the inertia of B is (n, 0, n), then there exists a hyperboic Q(λ) = λ 2 A + λb + C and a nonsinguar matrix U C 2n 2n such that the foowing statements are true. [ ] x 1. If (µ, x) is a quadratic eigenpair of Q(λ), then (µ, U ) is an eigenpair of L(λ). µx ] [ ] ] [ x x [ x 2. If (µ, ) is an eigenpair of L(λ) and we define = U ỹ y 1, where x C ỹ n, then (µ, x) is a quadratic eigenpair of Q(λ) and y = µx. Proof. Since L(λ) is positive definite and the inertia of B is (n, 0, n), by Theorem A.1 there exists a nonsinguar matrix W such that W H A W = diag(λ +, Λ ) and W H BW = diag(i, I), where Λ + = diag(λ + 1,, λ+ n ), Λ = diag(λ 1,, λ n ) and λ ± i R and λ + i > λ j for a i and j. Set A = I, B = (Λ + + Λ ), C = Λ + Λ, [ ] [ Λ I (Λ+ Λ S = ) 1/2 ] 0 I 0 (Λ + Λ ) 1/2, Λ + and Q(λ) = λ 2 A + λb + C. It can be verified that corresponding to this Q(λ), L Q (λ) of (2.5) satisfies L Q (λ) = S H W H L(λ)W S. Since L(λ) is positive definite, there is a λ 0 R such that L(λ 0 ) 0 which impies L Q (λ 0 ) 0 and thus Q(λ 0 ) 0 by (2.7). Consequenty, this Q(λ) is hyperboic by item 1 of Theorem 2.1. Finay take U = W S for items 1 and 2. 7

9 Theorem 2.4. Let Q(λ) = λ 2 A+λB+C be hyperboic. Then for any X C n m satisfying X H AX = I m, (X H BX) 2 4(X H CX) 0. (2.9) Proof. For any y C m with y 2 = 1, write x = Xy. We have y H [ (X H BX) 2 4(X H CX) ] y = (X H BXy) H (X H BXy) 4(Xy) H C(Xy) = y 2 2 X H BXy 2 2 4(Xy) H C(Xy) y H (X H AX)y (2.10) [ y H (X H BXy) ] 2 4(Xy) H C(Xy) (Xy) H A(Xy) (2.11) = (x H Bx) 2 4x H Cx x H Ax > 0, (2.12) where we have used y 2 = 1 and X H AX = I m for (2.10), and used the Cauchy- Bunyakovsky-Schwarz inequaity for (2.11). Therefore (X H BX) 2 4(X H CX) 0 by (2.12). Theorem 2.5. Let Q(λ) = λ 2 A + λb + C be a hyperboic quadratic matrix poynomia of order n, and denote by λ ± i its quadratic eigenvaues which are arranged as in (2.4). Set Λ + = diag(λ + 1,, λ+ n ), Λ = diag(λ 1,, λ n ). (2.13) Then there exists nonsinguar Z C 2n 2n of the form [ ] U+ U Z =, (2.14) U + Λ + U Λ where U +, U C n n are nonsinguar and Υ := U 1 + U (2.15) is unitary, such that [ ] Z H A Z = Z H C Z = A [ ] Z H BZ = Z H B A Z = A [ Λ+ [ In ], (2.16a) Λ ]. (2.16b) I n Write U + = [u + 1, u+ 2,..., u+ n ], U = [u 1, u 2,..., u n ]. As a consequence of (2.14) and (2.16), we have the foowing statements. 1. Q(λ + i )u+ i = 0, Q(λ i )u i = 0 for i = 1, 2,, n. Thus there are n ineary independent quadratic eigenvectors associated with a λ + i, and the same can be said about quadratic eigenvectors associated with a λ i. 2. ς(u ± i ) = 1 for i = 1, 2,..., n. 8

10 3. Q(λ) admits Q(λ) = U H )U H AU + (λi Λ + )U+ 1 (2.17a) Q(λ) = U+ H +)U+ H AU (λi Λ )U 1 (2.17b) 4. U H AU + = (Λ + Υ Υ Λ ) 1. As a resut, A, B, C and Q(λ) can be expressed in terms of Λ ± and any two of U +, U, and Υ, assuming (2.15). In particuar, where A = U+ H 1 ΘU+ (2.18a) B = U+ H + Λ + Θ)U+ 1 (2.18b) C = U+ H (Λ +ΘΛ + Λ + )U+ 1 [, ] (2.18c) Q(λ) = U+ H (λi Λ + )Θ(λI Λ + ) + (λi Λ + ) U+ 1, (2.18d) Θ = (Λ + Υ Λ Υ H ) 1. (2.18e) 5. We have U + 2 = U 2 A 1/2 2, (2.19a) λ + 1 λ n U+ 1 2 = U 1 2 A 1/2 2 λ + n λ 1, (2.19b) κ(u + ) = κ(u ) λ + n λ 1 κ(a) λ + 1, λ n (2.19c) and Z 2 Ξ U ± 2, Z 1 2 Ξ λ + 1 λ n U 1 ± 2, (2.20) where ξ ± = max{ λ ± 1, λ± n } and Ξ = 2 + ξ2 + + ξ 2 + [(ξ + 1) 2 + (ξ + 1) 2 ][(ξ + + 1) 2 + (ξ 1) 2 ]. 2 The foowing converse to item 4 is aso true: given diagona matrices Λ ± as in (2.13) and two of U +, U, and Υ, where Υ C n n as in (2.15) is unitary and U +, U C n n are nonsinguar, if λ ± i can be arranged as in (2.4), then the quadratic matrix poynomia constructed by (2.18) is hyperboic. Proof. Since Q(λ) is hyperboic, L Q (λ) in (2.5) is a positive definite penci. By Theorem A.1, there exists a nonsinguar Z C 2n 2n to give (2.16). We have to show that Z must take the form (2.14). Since each coumn of Z is an eigenvector of the penci L Q (λ), by Theorem 2.2, we [ ] [ u + concude that the ith coumn of Z can be expressed as i u ] λ + for 1 i n and j i u+ i λ j u j 9

11 for 1 j = i n n, where u + i, u j are the corresponding quadratic eigenvectors of Q(λ) associated with λ + i and λ j, respectivey. Hence Z takes the form (2.14). Bockwise, the equations in (2.16) yied U H + CU + Λ + U H + AU + Λ + = Λ +, (2.21a) U H CU Λ U H AU Λ = Λ, (2.21b) U H + CU Λ + U H + AU Λ = 0, (2.21c) U H + BU + + U H + AU + Λ + + Λ + U H + AU + = I, (2.21d) U H BU + U H AU Λ + Λ U H AU = I, (2.21e) U H + BU + U H + AU Λ + Λ + U H + AU = 0. (2.21f) We caim that U + is nonsinguar. Consider U + x = 0 for some x C n. We wi prove that x = 0 and thus U + is nonsinguar. By (2.21d), x H x = x H Ix = x H (U H + BU + + U H + AU + Λ + + Λ + U H + AU + )x = 0 which impies x = 0, as was to be shown. Simiary, U is nonsinguar. Next, we define Λ + := U + Λ + U+ 1, Λ := U Λ U 1. (2.22) We deduce from (2.21c) and (2.21f) the expressions for C and B in (2.23a) beow, and then use C = C H and B = B H to get (2.23b). C = Λ H A Λ +, B = A Λ + Λ H A, (2.23a) C = Λ H +A Λ, B = A Λ Λ H +A. (2.23b) Using the second equation in (2.23a), we deduce from (2.21d) and (2.21e) that So U+ H U + 1 = (U H U H + U 1 + = B + A Λ + + Λ H +A = ( Λ + Λ ) H A, U H U 1 = B A Λ Λ H A = A( Λ + Λ ). U 1 )H = U H U 1. Thus, (U 1 + U ) H U 1 + U = U H U H + U 1 + U = I, which infers Υ := U+ 1 U is unitary. Item 1 is straightforward. We now prove item 2 for u + i and the case for u i can be handed in exacty the same way. Write a i = (u + i )H Au + i, b i = (u + i )H Bu + i, and c i = (u + i )H Cu + i. By (2.21a) and (2.21d), we have soving which for c i and b i to get For item 3, we have, by (2.23), c i (λ + i )2 a i = λ + i, b i + 2a i λ + i = 1 b 2 i 4a i c i = (1 2a i λ + i )2 4a i [ λ + i + (λ + i )2 a i ] = 1. Q(λ) = (λi Λ H )A(λI Λ + ), Q(λ) = (λi Λ H +)A(λI Λ ) 10

12 which, together with (2.22), yied (2.17). For item 4, write Λ ;Υ = Υ Λ Υ H, then Λ + Λ ;Υ 0 because for x 0, which aso impies x H (Λ + Λ ;Υ )x λ + 1 xh x λ n x H Υ H Υ x = (λ + 1 λ n )x H x > 0 0 (Λ + Λ ;Υ ) 1 (λ + 1 λ n ) 1 I. (2.24) Substitute U = U + Υ into (2.21c) to get U+ H CU + Λ + U+ H AU + Λ ;Υ (2.21a), we have 0 = U H + CU + Λ + U H + AU + Λ + + Λ + = 0 and thus by = Λ + U+ H AU + Λ ;Υ Λ + U+ H AU + Λ + + Λ + ] = Λ + [I U+ H AU + (Λ + Λ ;Υ ). (2.25) Substitute U + = U Υ H into (2.21c) to get U H CU Λ +;Υ U H AU Λ = 0, where Λ +;Υ = Υ H Λ + Υ. Thus by (2.21b), we have 0 = U H CU Λ U H AU Λ Λ = Λ +;Υ U H AU Λ Λ U H AU Λ Λ ] = [I (Λ +;Υ Λ )U H AU Λ. (2.26) We note that at east one of Λ + and Λ is nonsinguar. If Λ + is nonsinguar, then (2.25) impies U H + AU + (Λ + Λ ;Υ ) = I U H + AU + = (Λ + Λ ;Υ ) 1. (2.27) If Λ is nonsinguar, then (2.26) impies (Λ +;Υ Λ )U H AU = I which, upon using U = U + Υ, aso impies the second equation in (2.27). Then U H AU + = (Λ + Υ Υ Λ ) 1. So, U H + AU + = Θ, U H + BU + = ΘΛ + Λ ;Υ Θ, and U H + CU + = Λ ;Υ ΘΛ +. Noticing Λ ;Υ Θ = (Λ + Λ ;Υ )Θ + Λ + Θ = I + Λ + Θ, we have (2.18). For item 5, the equaities in (2.19) is a consequence of U = U + Υ and that Υ is unitary. We now prove (2.19) for U +. Use (A 1/2 U + ) H (A 1/2 U + ) = Θ to get U + 2 A 1/2 2 A 1/2 U + 2 = A 1/2 2 Θ 2 A 1/2 2, λ + 1 λ n and use (U 1 + A 1/2 )(U 1 + A 1/2 ) H = Θ 1 to get U U 1 + A 1/2 2 A 1/2 2 = Θ 1 2 A 1/2 2 A 1/2 2 λ + n λ 1. They give (2.19a) and (2.19b) for U +. Combine (2.19a) and (2.19b) to get (2.19c). For the first inequaity in (2.20), we have [ [ Z 2 U+ 2 U = U U + 2 ξ + U 2 ξ + 2 = U ] ξ + ξ + 2 Ξ. ]

13 For the second inequaity, we notice by using U = U + Υ [ ] [ ] [ ] [ U+ 0 I Υ U+ 0 I 0 Z = = 0 U + Λ + Υ Λ 0 U + Λ + I ] [ I Υ 0 S where S = Υ Λ Λ + Υ = Θ 1 Υ. This expression, after some cacuations, eads to [ ] [ ] [ ] Z 1 I Υ S 1 I 0 U 1 = 0 S Λ + I 0 U+ 1 [ Υ S = 1 Υ Λ Υ H Υ S 1 ] [ ] U 1 S 1 Λ + S U+ 1, and thus Z 1 2 S 1 2 [ ξ 1 ξ + 1] 2 U = U Θ 2 Ξ which impies the second inequaity in (2.20). We now prove the converse of item 4. First Θ is Hermitian and Θ 0 by (2.24). Obviousy A, B, C in (2.18) is Hermitian and A 0. Choose λ 0 = (λ λ n )/2, then Θ 1 Λ + λ 0 I 0 and Θ (Λ + λ 0 I) 1. Thus, U H + Q(λ 0 )U + = (Λ + λ 0 I)Θ(Λ + λ 0 I) (Λ + λ 0 I) 0 which says Q(λ 0 ) 0. By item 1 of Theorem 2.1, Q(λ) is hyperboic. Remark Each of the decompositions in (2.17) doesn t refect the symmetry property in Q(λ) somewhat. However, using the fact that Υ = U+ 1 U is unitary, we can turn them into Q(λ) = U H + (λi Υ Λ Υ H )(Λ + Υ Λ Υ H ) 1 (λi Λ + )U 1 +, (2.28a) Q(λ) = U H (λi Υ H Λ + Υ )(Υ Λ + Υ H Λ ) 1 (λi Λ )U 1. ], (2.28b) These equations are essentiay the decomposition in [43, Theorem 31.24] but with more detai. 2. [22, Lemma 6.1] and Probem gen_hyper2 of [5] provide a different set of formuas for B and C: B = U+ H [ Θ(Λ 2 + Υ Λ 2 Υ H )Θ ] U+ 1, (2.29a) C = U+ H [ Θ(Λ 3 + Υ Λ 3 Υ H )Θ + Θ(Λ 2 + Υ Λ 2 Υ H )Θ(Λ 2 + Υ Λ 2 Υ H )Θ ] U+ 1. (2.29b) [31, Coroary 6] provides yet another formua for C: C = U+ H [ (Λ 1 + Υ Λ 1 Υ H ) 1] U+ 1. (2.30) Athough both (2.29) and (2.30) ook more compicated than ours for B and C in (2.18b) and (2.18c), they are actuay the same in theory. In fact, we have Θ(Λ 2 + Υ Λ 2 Υ H )Θ = Θ(Λ 2 + [Λ + Θ 1 ] 2 )Θ 12

14 which says (2.29a) is the same as (2.18b). = Λ + Θ + ΘΛ + I (2.31) Λ 1 + Υ Λ 1 Υ H = Λ 1 + [Λ + Θ 1 ] 1 (use (2.18e)) = Λ 1 + ( Θ 1 )[Λ + Θ 1 ] 1 (use X 1 Y 1 = X 1 [Y X]Y 1 ) = (Λ + ΘΛ + Λ + ) 1. So (2.30) is the same as (2.18c). Finay Θ(Λ 3 + Υ Λ 3 Υ H )Θ = Θ(Λ 3 + [Λ + Θ 1 ] 3 )Θ Therefore use aso (2.31) to get = Θ 1 + ΘΛ Λ 2 +Θ + ΘΛ + Θ 1 Λ + Θ ΘΛ + Θ 1 Θ 1 Λ + Θ Λ +. Θ(Λ 3 + Υ Λ 3 Υ H )Θ + Θ(Λ 2 + Υ Λ 2 Υ H )Θ(Λ 2 + Υ Λ 2 Υ H )Θ = (Θ 1 + ΘΛ Λ 2 +Θ + ΘΛ + Θ 1 Λ + Θ ΘΛ + Θ 1 Θ 1 Λ + Θ Λ + ) + (ΘΛ + + Λ + Θ I)Θ 1 (ΘΛ + + Λ + Θ I) = (Θ 1 + ΘΛ Λ 2 +Θ + ΘΛ + Θ 1 Λ + Θ ΘΛ + Θ 1 Θ 1 Λ + Θ Λ + ) + Θ 1 ΘΛ + Θ 1 Λ + Λ + + ΘΛ Λ + ΘΛ + Θ 1 Λ + Θ + ΘΛ + Θ 1 Λ + Θ + Λ 2 +Θ = Λ + + Λ + ΘΛ + which proves that (2.29b) is the same as (2.18c). 3. Λ± in (2.22) are two soutions of the matrix equation In fact, AX 2 + BX + C = 0. (2.32) A(U + Λ + U 1 + )2 + B(U + Λ + U 1 + ) + C = (AU +Λ BU + Λ + + CU + )U 1 + = 0, and simiary for A(U Λ U 1 )2 + B(U Λ U 1 ) + C = 0. On the other hand, the abiity of soving (2.32) factorizes Q(λ) into the product of two inear matrix poynomias, based on which Guo and Lancaster [20] proposed their sovent approach for soving HQEP (1.1) of modest sizes. 13

15 3 Variationa principes Throughout this section, Q(λ) = λ 2 A+λB+C C n n wi be aways assumed a hyperboic quadratic matrix poynomia and the notations in Theorem 2.5 wi be kept. The scaar λ 0 is as in item 1 of Theorem 2.1 such that Q(λ 0 ) 0. Consider the foowing equation in λ f(λ, x) := x H Q(λ)x = λ 2 (x H Ax) + λ(x H Bx) + (x H Cx) = 0, (3.1) given x 0. Since Q(λ) is hyperboic, this equation aways has two distinct rea roots (as functions of x) ρ ± (x) = xh Bx ± [ (x H Bx) 2 4(x H Ax)(x H Cx) ] 1/2 2(x H. (3.2) Ax) We sha ca ρ + (x) the pos-type Rayeigh quotient of Q(λ) at x, and ρ (x) the neg-type Rayeigh quotient of Q(λ) at x. It is easy to verify that for any x 0, ρ ± (x) R, and ρ ± (αx) = ρ ± (x) for any α C. By the eementary knowedge of scaar quadratic poynomias, we have ρ + (x) + ρ (x) = xh Bx x H Ax, ρ +(x) ρ (x) = xh Cx x H Ax. (3.3) Both wi be used ater in this paper. Theorem 3.1. We have ρ + (x) [λ + 1, λ+ n ], ρ (x) [λ 1, λ n ], (3.4) ς(x) := [ (x H Bx) 2 4(x H Ax)(x H Cx) ] 1/2 = ±[2ρ± (x) x H Ax + x H Bx], (3.5) ς 0 (x) = ς(x) x H x [(λ+ 1 λ n )λ min (A), (λ + n λ 1 )λ max(a)]. (3.6) Consequenty, λ + i = ρ + (u + i ) for the quadratic eigenpair (λ+ i, u+ i ) and ρ (u j ) = λ j for the quadratic eigenpair (λ j, u j ). Proof. By item 3 of Theorem 2.1, for any fixed nonzero x, f(λ, x) < 0 for λ (λ n, λ + 1 ) and f(λ, x) > 0 for λ (, λ 1 ) (λ+ n, + ). Thus, the arger root of the scaar quadratic equation f(λ, x) = 0 in λ must ie in [λ + 1, λ+ n ] and the smaer one in [λ 1, λ n ]. That is (3.4). For (3.5), we have 2ρ ± (x) x H Ax + x H Bx = [ x H Bx ± (x H Bx) 2 4(x H Ax)(x H Cx) ] + x H Bx = ±ς(x). Lasty, the incusion (3.6) is a resut of ς(x) = [ρ + (x) ρ (x)] x H Ax. 14

16 3.1 Courant-Fischer type min-max principes Theorem 3.2 beow is a restatement of [43, Theorem 32.10, Theorem and Remark 32.13]. However, it is essentiay due to Duffin [12, 1955] whose proof, athough for overdamped Q, works for the genera hyperboic case. Cosey reated ones for more genera noninear eigenvaue probems (other than quadratic eigenvaue probems) can be found in [49, 50, 66, 67]. They can be considered as a generaization of the Courant-Fischer min-max principes (see [47, p.206], [56, p.201]). Theorem 3.2 ([12]). We have for 1 i n λ + i = max X C n codim X=i 1 min x X x 0 ρ + (x), λ + i = min max ρ + (x), X C n x X dim X=i x 0 λ i = max X C n codim X=i 1 min x X x 0 ρ (x), λ i = min max ρ (x). X C n x X dim X=i x 0 (3.7a) (3.7b) (3.7c) (3.7d) In particuar, λ + 1 = min x 0 ρ +(x), λ 1 = min x 0 ρ (x), λ + n = max x 0 ρ +(x), λ n = max x 0 ρ (x). (3.8a) (3.8b) 3.2 Wieandt-Lidskii type min-max principes Theorems 3.3 and 3.4 which can be considered as generaizations of Amir-Moéz type minmax principes [1] and Theorem 3.5 which can be considered as generaizations of the Wieandt-Lidskii min-max principes ([39, 69] and aso [6, p.67], [56, p.199]) and Ky-Fan trace min/max principes [15] are new. For the ease of stating them, et λ ± R such that λ λ 1 λ n λ 0 λ + 1 λ+ n λ +. Such λ ± exist, e.g., λ = λ 1 or and λ + = λ + n or. Set intervas { { I + = [λ 0, λ + ], if λ + <, [λ, λ 0 ], if λ >, I = [λ 0, ), otherwise, (, λ 0 ], otherwise. (3.9) The foowing emma is aso essentiay due to Duffin [12] whose proof, athough for overdamped Q, again works for the genera hyperboic case. Lemma 3.1. We have λ + i ρ + (x) for any x span{u + 1, u+ 2,..., u+ i }, (3.10a) λ + i ρ + (x) for any x span{u + i, u+ i+1,..., u+ n }. (3.10b) 15

17 To generaize Amir-Moéz/Wieandt-Lidskii min-max principes, we introduce the foowing notations. For X C n k with rank(x) = k, X H Q(λ)X is a k k hyperboic quadratic matrix poynomia. Hence its quadratic eigenvaues are rea. Denote them by λ ± i,x arranged as λ 1,X λ k,x λ+ 1,X λ+ k,x. (3.11) Theorem 3.3. Let 1 i 1 < < i k n. For any Φ : I + I }{{ + R } k that is non-decreasing in each of its arguments, we have 2 min X 1 X k dim X j =i j max X 1 X k codim X j =i j 1 sup x j X j, j=1,...,k X=[x 1,...,x k ] rank(x)=k inf x j X j, j=1,...,k X=[x 1,...,x k ] rank(x)=k Φ(λ + 1,X,, λ+ k,x ) = Φ(λ+ i 1,, λ + i k ), Φ(λ + 1,X,, λ+ k,x ) = Φ(λ+ i 1,, λ + i k ). (3.12a) (3.12b) If aso Φ is continuous, then sup in (3.12a) and inf in (3.12b) can be repaced by max and min, respectivey. In particuar, setting i j = j in (3.12a) and setting i j = j + n k in (3.12b), respectivey, give min rank(x)=k Φ(λ+ 1,X,, λ+ k,x ) = Φ(λ+ 1,, λ+ k ), max rank(x)=k Φ(λ+ 1,X,, λ+ k,x ) = Φ(λ+ n k+1,, λ+ n ). Proof. For convenience, we define, for a matrix W = [w 1,..., w p ], (3.13a) (3.13b) S j,w := span{w 1,, w j }, T j,w := span{w j,, w p } for j = 1,, p. In particuar S W = S p,w, T W = T 1,W, and thus S W = T W. First we prove (3.12b). Reca the quadratic eigenvectors u + j introduced in Theorem 2.5. Choose X j = span{u + i j,, u + n } for j = 1, 2,..., k. (3.14) Then X 1 X k and codim X j = i j 1. By Lemma 3.1, ρ + (x) λ + i j for any x X j. Therefore min ρ + (x) = λ + i j. x X j x 0 For any X = [x 1,..., x k ] with x j X j for j = 1,, k such that rank(x) = k, consider X H Q(λ)X which is a k k hyperboic quadratic matrix poynomia. For j = 1,, k, noticing T j,x X j, we have by Theorem 3.2 λ + j,x = max X T X dim X=k j+1 min ρ + (x) min ρ + (x) min ρ + (x) = λ + i x X x T j. j,x x X x 0 j x 0 x 0 2 In (3.12a), it is not cear if the sup is attainabe for any given X j satisfying the given assumptions, except for continuous Φ. The same comment appies to the inf in (3.12b). 16

18 Since Φ( ) is non-decreasing in each of its arguments, Φ(λ + 1,X,, λ+ k,x ) Φ(λ+ i 1,, λ + i k ) which gives min x j X j, j=1,...,k X=[x 1,...,x k ] rank(x)=k Φ(λ + 1,X,, λ+ k,x ) Φ(λ+ i 1,, λ + i k ) because x j X j for 1 i k are arbitrary, subject to rank(x) = k. Therefore sup X 1 X k codim X j =i j 1 inf x j X j, j=1,...,k X=[x 1,...,x k ] rank(x)=k Φ(λ + 1,X,, λ+ k,x ) Φ(λ+ i 1,, λ + i k ). (3.15) On the other hand, et X j for j = 1,, k be any subspaces that satisfy the assumptions: X 1 X k and codim X j = i j 1. Define Y j = span{u + 1,, u+ i j }. Then Y 1 Y k and dim Y j = i j. By [1, Coroary 2.2] (see aso [37, Lemma 3.2]), there exists two A-orthonorma sets {x 1,..., x k } and {y 1,, y k } with x j X j for j = 1,..., k and y j Y j for 1 j k such that T X := span{x 1,, x k } = span{y 1,, y k } =: S Y. where X = [x 1,..., x k ] and Y = [y 1,, y k ] satisfy X H AX = Y H AY = I k. Y H Q(λ)Y is a hyperboic quadratic matrix poynomia whose pos-type quadratic eigenvaues are λ + 1,Y λ+ k,y. Since S Y = T X, λ + j,y = λ+ j,x for j = 1,, k. By Lemma 3.1, ρ + (y) λ + i j for any y Y j. Therefore max ρ + (y) = λ + i y Y j. j y 0 By Theorem 3.2 and noticing S j,y Y j, we have, for j = 1,, k, λ + j,x = λ+ j,y = min Y S Y dim Y=j max ρ + (y) max y Y y S j,y y 0 y 0 Since Φ( ) is non-decreasing in each of its arguments, which gives inf x j X j, j=1,...,k X=[x 1,...,x k ] rank(x)=k Since X j are arbitrary, we concude sup X 1 X k codim X j =i j 1 Φ(λ + 1,X,, λ+ k,x ) Φ(λ+ i 1,, λ + i k ), inf ρ + (y) max ρ + (y) = λ + i y Y j. j y 0 Φ(λ + 1,X,, λ+ k,x ) Φ(λ+ i 1,, λ + i k ). x j X j, j=1,...,k X=[x 1,...,x k ] rank(x)=k Φ(λ + 1,X,, λ+ k,x ) Φ(λ+ i 1,, λ + i k ). (3.16) 17

19 Combine (3.15) and (3.16) to get sup X 1 X k codim X j =i j 1 inf x j X j, j=1,...,k X=[x 1,...,x k ] rank(x)=k Φ(λ + 1,X,, λ+ k,x ) = Φ(λ+ i 1,, λ + i k ). (3.12b ) But the sup here is achievabe by the seection in (3.14). Thus we have (3.12b). Now we caim the inf can be repaced by min for a continuous Φ. Let X j for j = 1,, k be given and satisfy the assumptions: X 1 X k and codim X j = i j 1. There exist a sequence X (i) C n k with rank(x (i) ) = k and its jth coumn in X j such that im i Φ(λ+,, λ + ) = 1,X (i) k,x (i) inf x j X j, j=1,...,k X=[x 1,...,x k ] rank(x)=k Φ(λ + 1,X,, λ+ k,x ). (3.17) Without oss of generaity, we may assume X (i) has A-orthonorma coumns, i.e., (X (i) ) H AX (i) = I k ; otherwise we can perform the Gram-Schimdt A-orthogonaization on the coumns of X (i) from the ast coumn backwards, and the new X (i) has the same property as the od X (i) : rank(x (i) ) = k and its jth coumn in X j, and aso λ ± j,x (i) remain the same. Since {X (i) } is a bounded set in C n k, it has a convergent subsequence. Through renaming, we may assume that {X (i) } itsef is convergent, and et Y C n k be the imit. It is not hard to see that Y H AY = I k which impies rank(y ) = k and that the jth coumn of Y is in X j. Since (X (i) ) H Q(λ)X (i) goes to Y H Q(λ)Y, by the continuity of quadratic eigenvaues with respect to the coefficient matrices we concude im i λ± = λ ± j,x (i) j,y for 1 j k. Therefore the eft-hand side of (3.17) is equa to Φ(λ + 1,Y,, λ+ k,y ), and thus the inf in (3.17) is attainabe. For (3.12a), a proof simiar to what we did above for (3.12b) works: choosing X j = span{u + 1,, u+ i j } wi ead to that the eft-hand side is no bigger than its right-hand side, and choosing Y j = span{u + i j,, u + n } wi give the opposite. Simiary to Theorem 3.3, we have Theorem 3.4. Let 1 i 1 < < i k n. For any Ψ : I I }{{ R } k that is non-decreasing in each of its arguments, we have 3 min X 1 X k dim X j =i j sup x j X j, j=1,...,k X=[x 1,...,x k ] rank(x)=k Ψ(λ 1,X,, λ k,x ) = Ψ(λ i 1,, λ i k ), (3.18a) 3 In (3.18a), it is not cear if the sup is attainabe for any given X j satisfying the given assumptions. The same comment appies to the inf in (3.18b). 18

20 max X 1 X k codim X j =i j 1 inf x j X j, j=1,...,k X=[x 1,...,x k ] rank(x)=k Ψ(λ 1,X,, λ k,x ) = Ψ(λ i 1,, λ i k ). (3.18b) If aso Ψ is continuous, then sup in (3.18a) and inf in (3.18b) can be repaced by max and min, respectivey. In particuar, setting i j = j in (3.18a) and setting i j = j + n k in (3.18b), respectivey, give min rank(x)=k Ψ(λ 1,X,, λ k,x ) = Ψ(λ 1,, λ k ), max rank(x)=k Ψ(λ 1,X,, λ k,x ) = Ψ(λ n k+1,, λ n ). (3.19a) (3.19b) Proof. Consider the hyperboic quadratic matrix poynomia Q(λ) = λ 2 A + λ( B) + C whose quadratic eigenvaues are ˆλ 1 ˆλ n < ˆλ + 1 ˆλ + n, where ˆλ i = λ + n i+1 and ˆλ + j = λ n j+1. Appy (3.12b) to Q(λ) with Φ(ξ 1,..., ξ k ) := Ψ( ξ k,..., ξ 1 ) to get (3.18a), and appy (3.12a) to Q(λ) with the same Φ to get (3.18b). Speciaizing Theorems 3.3 and 3.4 to the case where Φ and Ψ are the sum of its arguments gives us Wieandt-Lidskii type min-max principes as summarized in the foowing theorem and Ky-Fan type trace min/max principes. Theorem 3.5. Let 1 i 1 < < i k n and typ {+, }. Then min X 1 X k dim X j =i j max X 1 X k codim X j =i j 1 max x j X j X=[x 1,...,x k ] rank(x)=k min x j X j X=[x 1,...,x k ] rank(x)=k j=1 j=1 λ typ j,x = λ typ j,x = j=1 j=1 λ typ i j, (3.20a) λ typ i j. (3.20b) In particuar, setting i j = j in (3.20a) and setting i j = j + n k in (3.20b) give min rank(x)=k j=1 λ typ j,x = j=1 λ typ j, max rank(x)=k 3.3 Cauchy type interacing inequaities j=1 λ typ j,x = j=1 λ typ n k+j. (3.21) The Cauchy type interacing inequaities in (3.22) were recenty obtained by Veseić [64]. Here we present a simpe proof, using our generaizations of Amir-Moéz type min-max principes in Theorems 3.3 and

21 Theorem 3.6 (Cauchy-type interacing inequaities [64]). Suppose X C n k with rank(x) = k. Denote the quadratic eigenvaues of X H Q(λ)X by Then Proof. Let µ 1 µ k < µ+ 1 µ+ k. λ + i µ + i λ + i+n k, i = 1,, k, (3.22a) λ j µ j λ j+n k, j = 1,, k. (3.22b) Φ(α 1,, α k ) = the ith argest α j. Then this Φ satisfies the condition of Theorem 3.3. Making use of (3.13a) and (3.13b) gives µ + i λ + i and µ + i λ + i+n k, respectivey. That is (3.22a). Simiary, we get (3.22b) by Theorem 3.4. Remark 3.1. The Cauchy type interacing inequaities in Theorem 3.6 are sharper than those possiby derivabe by inearization. Actuay, through inearization and by item 1 of [38, Theorem 1.1] (which is, in fact, [30, Theorem 2.1]), we can ony obtain λ + i µ + i λ + i+2n 2k, i = 1,, k, λ j (n k) µ j λ j+n k, j = 1,, k, where we set λ + i = + for i > n and λ j = for j < 1. 20

22 4 Perturbation anaysis 4.1 Setting the stage Throughout this section, we suppose that Hermitian matrices A, B, and C are perturbed to Hermitian matrices Ã, B, and C and set A = à A, B = B B, C = C C. (4.1) This notationa convention of pacing a over a symbo for the corresponding perturbed quantity and a before a symbo for the change in the quantity wi be generaized to a quantities that depend on A, B, and C. For exampe, Q(λ) = λ 2 A + λb + C is perturbed to Q(λ) = λ 2 à + λ B + C, as a resut, and ρ ± (x) = (x H Bx) ± [ (x H Bx) 2 4(x H Ãx)(x H Cx) ] 1/2 2(x HÃx) (xh Bx) ± [ (x H Bx) 2 4(x H Ax)(x H Cx) ] 1/2 2(x H. Ax) Besides A 0, the other key condition for Q(λ) = λ 2 A + λb + C to be hyperboic is [ς(x)] 2 = (x H Bx) 2 4(x H Ax)(x H Cx) > 0, for a 0 x C n. (2.2) We first estabish a condition under which (2.2) is weaky 4 satisfied for a convex combination (1 t) Q(λ) + t Q(λ). To this end, we define ϕ(x) := (x H Bx) 2 4(x H Ax)(x H Cx), (4.2) ψ(x) := (x H Bx)(x H Bx) 2(x H Ax)(x H Cx) 2(x H Cx)(x H Ax), (4.3) and define ϕ(x) and ψ(x) in the same way, except by swapping the positions of A, B, C with those of Ã, B, and C. It can be verified that ϕ(x) = ϕ(x), ψ(x) = ψ(x) ϕ(x). Aso define g(t) : = (x H [B + t B]x) 2 4(x H [A + t A]x)(x H [C + t C]x) = ς(x) 2 + 2ψ(x)t + ϕ(x)t 2. So g(0) = ς(x) and g(1) = ς(x). Correspondingy, g(t) : = (x H [ B t B]x) 2 4(x H [à t A]x)(xH [ C t C]x) = ς(x) ψ(x)t + ϕ(x)t 2. Note that g(t) = g(1 t). By definition, if A 0, then Q(λ) is hyperboic if and ony if g(0) > 0 for any nonzero x C n, and if à 0, then Q(λ) is hyperboic if and ony if g(1) > 0 for any nonzero x C n. 4 By weaky, we mean the strict positivity in (2.2) is given in to nonnegativity. 21

23 Lemma 4.1. Suppose min{g(0), g(1)} 0. Then g(t) 0 for a 0 t 1 and nonzero x C n if and ony if min{ϕ(x), ψ(x), ψ(x), ψ(x) 2 ϕ(x)ς(x) 2 } 0 for a x 0. (4.4) Proof. The condition (4.4) is equivaent to that for any x, at east one of ϕ(x) 0, ψ(x) 0, ψ(x) = ψ(x) ϕ(x) 0, ψ(x) 2 ϕ(x)ς(x) 2 0 hods. Note that g(0) 0 and g(1) 0 by assumption. We first prove that (4.4) impies g(t) 0 for a 0 t 1 and for any nonzero x C n. To this end, we et 0 t 1 and 0 x C n. 1. If ϕ(x) 0, then g(t) is concave and thus g(t) (1 t)g(0) + tg(1) 0; 2. If ψ(x) 0, then g(t) = ς(x) 2 + 2ψ(x)t + ϕ(x)t 2 ς(x) 2 + 2ψ(x)t 2 + ϕ(x)t 2 = (1 t 2 )g(0) + t 2 g(1) 0; 3. If ψ(x) 0, then simiary g(t) (1 t 2 ) g(0) + t 2 g(1) 0; 4. Consider the case ψ(x) 2 ϕ(x)ς(x) 2 0. Suppose 5 ϕ(x) > 0. Then g(t) is a nontrivia quadratic function and has at most one zero in R. Going through the cases either there is no zero or the zero is in (0, 1) or the zero is outside of (0, 1), we can see g(t) 0 for a 0 t 1. Next for the necessity of (4.4), suppose there were an x 0 satisfying ϕ(x) > 0, ψ(x) < 0, ψ(x) = ψ(x) + ϕ(x) > 0, and ψ(x) 2 ϕ(x)ς(x) 2 > 0. Then min t g(t) = ψ(x)2 ϕ(x)ς(x) 2 ϕ(x) < 0 and min t g(t) is attained at t min = ψ(x) ϕ(x) g(t) 0 for 0 t 1. (0, 1), contradicting the assumption that where Given a shift λ 0 R, define Q λ0 (λ) := Q(λ + λ 0 ) = λ 2 A + λ(2λ 0 A + B) + Q(λ 0 ) (4.5) = λ 2 A + λb λ0 + C λ0, (4.6) B λ0 = 2λ 0 A + B, C λ0 = Q(λ 0 ). (4.7) It can be verified that (µ, x) is a quadratic eigenpair of Q λ0 (λ) if and ony if (µ + λ 0, x) is a quadratic eigenpair of Q(λ). 5 The case ϕ(x) 0 has aready been deat with. 22

24 Lemma 4.2. Suppose that Q(λ) is hyperboic, and adopt the notations introduced in Theorem If λ 0 (λ n, λ + 1 ), then diag( C λ 0, A) = diag( Q(λ 0 ), A) If λ 0 [λ + n, + ), then Q λ0 (λ) is overdamped, i.e. B λ0 0 and C λ0 0. Moreover, (λ n + λ + n 2λ 0 )A B λ0 (λ 1 + λ+ 1 2λ 0)A, (4.8) (λ n λ 0 )(λ + n λ 0 )A C λ0 (λ 1 λ 0)(λ + 1 λ 0)A. (4.9) 3. If A 1/2 AA 1/2 2 < 1, then à 0. Proof. Item 1 is a consequence of Theorem 2.1 and (4.7). For (4.8) of item 2, we have for any x 0 x H B λ0 x = 2λ 0 x H Ax + x H Bx ( ) = x H Ax 2λ 0 + xh Bx x H Ax = x H Ax ( 2λ 0 [ρ + (x) + ρ (x)] ) which, together with (3.4), yieds (4.8). For (4.9), we have for any x 0 x H C λ0 x = x H Q(λ 0 )x = x H Ax[λ 0 ρ + (x)][λ 0 ρ (x)] which, together with (3.4), yieds (4.9). For item 3, we notice the smaest eigenvaue of A 1/2 ÃA 1/2 satisfies λ min (A 1/2 ÃA 1/2 ) = 1 + λ min (A 1/2 AA 1/2 ) 1 A 1/2 AA 1/2 2 > 0 if A 1/2 AA 1/2 2 < Asymptotica anaysis It is a common technique to perform an asymptotica anaysis in numerica anaysis for at east three reasons: 1. it is mathematicay sound, provided it is known that the interested quantities are continuous with respect to what is being perturbed; 2. it is reativey easy because it is a first order anaysis, and 3. it is powerfu in reveaing the intrinsic sensitivity of the interested quantities. Let (µ, x) is a simpe quadratic eigenpair of HQEP (1.1) for Q(λ). Since HQEP (1.1) is equivaent to the eigenvaue probem for the reguar matrix penci L Q (λ) in (2.5) and since the eigenvaues of a reguar matrix penci and the eigenvectors associated with simpe eigenvaues are continuous with respect to the entries of the invoved matrices [56], Q(λ) has a simpe quadratic eigenpair ( µ, x) = (µ+ µ, x+ x) such that µ 0 and x 0 as A, B, C 0. Now suppose that A, B, and C are sufficienty tiny, 23

25 and so are µ and x. Ignoring terms of order 2 or higher and noticing Q(µ)x = 0, we have from Q(µ + µ) (x + x) = 0 µ [ 2µA + B ] x + [ µ 2 A + µ B + C ] x + [ µ 2 A + µb + C ] x 0, (4.10) where the means the equation is true after ignoring terms of order 2 or higher. Premutipy (4.10) by x H and use x H Q(µ) = 0 to get µ xh[ µ 2 A + µ B + C ] x x H[ 2µA + B ] x = xh[ µ 2 A + µ B + C ] x ς(x) (4.11) (4.12) = µ2 ±ς(x) xh Ax µ ±ς(x) xh Bx 1 ±ς(x) xh Cx. (4.13) where the equaity in (4.12) is due to (3.5). There is a cear interpretation of (4.13): the change µ is proportiona to A, B, C with mutipying factors µ 2 /ς(x), µ/ς(x), and 1/ ς(x), respectivey. Our foowing strict bounds refect this interpretation. The expression (4.11) is not new and its derivation foows a rather standard technique (see, e.g., [62]). What is new here is the use of (3.5) to reate its denominator x H[ 2µA+B ] x to ς(x), a quantity that determines the hyperboicity of Q. 4.3 Perturbation bounds in the spectra norm Throughout the rest of this section, we assume Q(λ) and Q(λ) are hyperboic and A 1/2 AA 1/2 2 < 1 (4.14) which guarantees à 0. We wi adopt the notations introduced in Theorem 2.5. Our goa is to bound the norms of Λ + = diag( λ + 1 λ+ 1,..., λ + n λ + n ), Λ = diag( λ 1 λ 1,..., λ n λ n ). Bounds on norms of the change to Λ = diag(λ, Λ + ) are easiy derivabe through Λ 2 = max Λ ± 2, Λ F = Λ + 2 ± F + Λ 2 F, Λ ui 2 max ± Λ ± ui. In this subsection, we wi focus on the spectra norm, and eave the case for the Frobenius norms and more generay unitariy invariant norms to the next subsection. Our main resuts of this subsection are summarized in Theorem 4.1. Theorem 4.1. Let typ {+, }, and ϵ a = A 1/2 AA 1/2 2, ϵ b = B 2 B 2, ϵ c = C 2 C 2, (4.15) λ typ max = max{ λ typ 1, λ typ n }, λtyp max = max{ λ typ 1, λ typ n }, (4.16) χ ς = min x 0 {ς 0(x), ς 0 (x)}, χ λ typ = max{λ typ max, λ typ max}. (4.17) 24

26 1. If A = B = 0 and then 2. If B = C = 0 and then 3. If A = C = 0 and ϵ c < χ 2 ς 4 A 2 C 2, (4.18) Λ typ 2 1 χ ς C 2. (4.19) { ϵ a < min 1, Λ typ 2 χ 2 ς 4 A 2 C 2 }, (4.20) χ2 λ typ (1 ϵ a )χ ς A 2. (4.21) then 4. If A = C = 0 and ϵ b < χ 2 ς B 2 ( B A 2 C 2 ), (4.22) Λ typ 2 χ λ typ B 2 + C 2 χ ς χ 3 B 2 2. (4.23) ς B 2 < χ 2 ς 2λ 0 A + B A 2 Q(λ 0 ) 2, (4.24) where λ 0 (, min{λ 1, λ 1 }] [max{λ+ n, λ + n }, + ), then Λ typ 2 χ λ typ + λ 0 χ ς B 2. (4.25) 5. In genera, without assuming two of A, B, and C are zeros, if { χ 2 } ς ϵ a < γ min 1,, (4.26a) 4 A 2 C 2 χ 2 ς ϵ b < γ B 2 ( B A 2 C 2 ), (4.26b) where γ = χ 2 ς ϵ c < γ, 4 A 2 C 2 χ 2 ς B χ2 ς + ( B χ2 ς )( B χ2 ς ) (4.26c) < 2 1, (4.27) then 4 [ Λ typ 2 (1 ϵ a )χ 3 C 2 A 2 C 2 (ϵ a + ϵ c ) 2 + B 2 2(ϵ b + ϵ a )(ϵ b + ϵ c ) ] ς 1 [ + (χλ typ) 2 ] A 2 + χ (1 ϵ a )χ λ typ B 2 + C 2. (4.28) ς 25

27 A bounds by this theorem are strict. They are consistent with the asymptotic expression (4.13) rather we after dropping terms of order 2 or higher in ϵ a, ϵ b, and ϵ c. For exampe, (4.28) yieds Λ typ 2 1 χ ς [ (χλ typ) 2 A 2 + χ λ typ B 2 + C 2 ]. (4.29) The rest of this subsection is devoted for the proof of Theorem 4.1. Later in the next subsection, we wi extend (4.19) to a genera unitariy invariant norm. Each of many expressions beow is in its compact form for two. For exampe, (4.30) incudes two dispayed equations: one for ρ + and one for ρ + with a ± seected as either + or, accordingy. Lemma 4.3. If (4.4) and (4.14) hod, then there exists 0 ξ 1 such that [ ] ρ ± (x) = δ ± (x, ξ) := ± δ 3 (x, ξ) xh Ax x HÃx δ± 2 (x) for any x 0, where (4.30) δ 2 ± (x) = ρ ±(x) 2 (x H Ax) + ρ ± (x)(x H Bx) + x H Cx, ς(x) (4.31a) ς(x) 2 ϕ(x) ψ(x) 2 δ 3 (x, ξ) = 4(x HÃx), [ς(x)2 + 2ψ(x)ξ + ϕ(x)ξ 2 3/2 ] (4.31b) ϕ(x) and ψ(x) are defined in (4.2) and (4.3). In addition, A 1/2 AA 1/2 xh Ax 2 x HÃx 1 1 A 1/2 AA 1/2, (4.32) 2 δ 2 ± (x) max{ λ± 1 2, λ ± n 2 } A 2 + max{ λ ± 1, λ± n } B 2 + C 2 min ς. (4.33) 0(x) x 0 Proof. According to how the difference operator is defined at the beginning of subsection 4.1, we have ± ρ ± (x) = ς(x) xh Bx 2(x H Ax) The rest of this proof is to cacuate ϵ 1 and ϵ 2. By Lemma 4.1, + ς(x) ( ) xh Bx 1 2 x H =: ϵ 1 + ϵ 2. (4.34) Ax f(t; x) := [ ς(x) 2 + 2ψ(x)t + ϕ(x)t 2] 1/2 (4.35) is we-defined and differentiabe for 0 t 1. By the Tayor expansion, there exists 0 ξ 1 such that ς(x) = f(1; x) = f(0; x) + f (0; x) f (ξ; x) = ς(x) + ψ(x) ς(x) + ς(x)2 ϕ(x) ψ(x) 2 2[f(ξ; x)] 3. (4.36) 26

28 This ξ depends on x. Now we are ready to cacuate ϵ 1 and ϵ 2. We have ( ϵ 1 = xh Bx 2(x H Ax) + 1 ψ(x) 2(x H Ax) ς(x) + ς(x)2 ϕ(x) ψ(x) 2 ) 2[f(ξ; x)] 3 and = xh Bx 2(x H Ax) + (xh Bx)(x H Bx) 2(x H xh Cx Ax)ς(x) ς(x) = ±ς(x) (xh Bx) x H Bx 2(x H xh Cx Ax) ς(x) ς(x) = ρ ±(x)(x H Bx) ς(x) xh Cx ς(x) = δ ± 2 (x) + ρ ±(x) 2 (x H Ax) ς(x) ϵ 2 = [ ς(x) xh Bx](x H Ax) 2(x HÃx)(xH Ax) Noticing we have xh Cx x H Ax ς(x) x H Ax + ς(x)2 ϕ(x) ψ(x) 2 4(x H Ax)[f(ξ; x)] 3 xh Cx x H Ax ς(x) x H Ax + ς(x)2 ϕ(x) ψ(x) 2 4(x H Ax)[f(ξ; x)] 3 xh Cx x H Ax ς(x) x H Ax + xh Ãx x H Ax xh Cx x H Ax ς(x) x H Ax + xh Ãx x H Ax δ 3(x, ξ) = ρ ±(x)(x H Ax) x H Ax x H Cx ς(x) ± ρ ±(x) = xh Cx ς(x) ± xh Bx ± ς(x) 2(x H Ax) = 2(xH Ax)(x H Cx) x H Bxς(x) + ς(x) 2 2ς(x)(x H Ax) ς(x) 2 ϕ(x) ψ(x) 2 4(xHÃx)[f(ξ; x)]3 = [±ρ ± (x) ± ρ ± (x)] xh Ax x H Ax. = (xh Bx) 2 ς(x) 2 2(x H Bx)ς(x) + 2ς(x) 2 4ς(x)(x H Ax) = [xh Bx ς(x)] 2 4ς(x)(x H Ax) = ρ ±(x) 2 (x H Ax), ς(x) ± ρ ± (x) = ϵ 1 + ϵ 2 = δ 2 ± (x) + xh Ãx x H Ax δ 3(x, ξ) [± ρ ± (x)] xh Ax x H Ax soving which for ± ρ ± (x) eads to ρ ± (x) = δ ± (x, ξ). Lemma 4.4. Suppose (4.4) and (4.14) hod. functions satisfying Let δ ± b (x), δ± ub (x), δ ± b (x), and δ ± ub (x) be δ ± b (x) δ± (x, ξ) δ ± ub (x), δ± b (x) δ ± (x, ξ) δ ± ub (x) (4.37) for a x C n, ξ [0, 1], where δ ± (x, ξ) is defined as in Lemma 4.3. Write Then γ ± uu = max x 0 { δ± ub (x), δ ± ub (x)}, γ± = max x 0 { δ± b (x), δ ± b (x)}, γ ± u = max x 0 { δ± b (x), δ± ub (x)}, γ± u = max x 0 { δ ± b (x), δ± ub (x)}. Λ ± 2 = max 1 i n λ± i min{γ± uu, γ ±, γ± u, γ± u }. (4.38) 27

29 Proof. We ony consider the + case beow; the case is simiar. repacing + with gives a proof for the case. By Lemma 4.3, δ + b (x) ρ +(x) = δ + (x, ξ) δ + ub (x). In fact simpy Let S i = span{u + 1,, u+ i }, T i = span{u + i,, u+ n } and simiary define S i and T i. By the Courant-Fischer type min-max principes in Theorem 3.2, Therefore, λ + i = min max ρ +(x) = max ρ + (x) = ρ + (u + i ), dim X=i 0 x X 0 x S i λ + i = min max ρ +(x) = max ρ + (x) = ρ + (ũ + i ), dim X=i 0 x X 0 x S i λ + i = max codim X=i 1 λ + i = max codim X=i 1 min ρ +(x) = 0 x X min ρ +(x) = 0 x X λ + i = min dim X=i max 0 x X ρ +(x) max 0 x S i λ + i = max codim X=i 1 min ρ +(x) 0 x X min ρ + (x) = ρ + (u + i ), 0 x T i min ρ + (x) = ρ + (ũ + i ). 0 x T i ρ + (x) max 0 x S i [ ρ+ (x) + δ + ub (x)] max 0 x S i ρ + (x) + max 0 x S i δ + ub (x) = λ + i + max 0 x S i δ + ub (x), min 0 x T i ρ + (x) min 0 x T i [ ρ+ (x) + δ + b (x)] min ρ + (x) + min δ + 0 x T i 0 x T b (x) i = λ + i + min 0 x T i δ + b (x). They give (4.39a) beow and (4.39b) as we, by switching the roes of Q and Q: min 0 x T i δ + b (x) λ + i λ + i max 0 x S i δ + ub (x), min 0 x T i δ+ b (x) λ+ i λ + i max 0 x S i δ+ ub (x). (4.39a) (4.39b) It foows from (4.39) that { λ + i max max δ + 0 x S ub (x), max δ+ ub (x) i 0 x S i max x 0 {δ+ ub (x), δ + ub (x)} = γ+ uu, λ + i { max min δ + 0 x T b (x), min δ+ b (x) i 0 x T i 28 } }

30 max x 0 { δ+ b (x), δ + b (x)} = γ+, λ + i { max min δ + 0 x T b (x), max δ + i 0 x S ub (x) i max x 0 { δ+ b (x), δ+ ub (x)} = γ+ u, λ + i max { min δ+ b 0 x T i (x), max 0 x S i δ+ max x 0 { δ + b (x), δ + ub (x)} = γ+ u. This competes the proof of (4.38) for the + case. ub (x) Proof of Theorem 4.1. We ony prove the perturbation resuts for Λ +. The case for Λ can be turned into one for Λ + by considering the pos-type quadratic eigenvaues of Q( λ) and Q( λ). For any α > 0, x 0, we have ϵ a < α x H Ax < αx H Ax, (4.40a) χ 2 ς ϵ a < α x H Ax < α ς(x)2 4 A 2 C 2 4 x H Cx, (4.40b) χ 2 ς ϵ c < α x H Cx < α ς(x)2 4 A 2 C 2 4x H Ax, } } (4.40c) χ 2 ς ϵ b < α B 2 ( B A 2 C 2 ) x H Bx < α x H Bx, (4.40d) where (4.40a) and (4.40b) hod because x H Ax x H Ax = x H A 1/2 (A 1/2 AA 1/2 )A 1/2 x x H A 1/2 A 1/2 x A 1/2 AA 1/2 2 = ϵ a, and (4.40d) hods because the eft part tes x H ς(x) 2 ( ) Bx < α x H Bx + 4(x H Ax) x H Cx = α x H Bx 4(x H Ax) x H Cx. (4.41) For item 1: A = B = 0, ϕ(x) = ϕ(x) = 0, ψ(x) = 2(x H Ax)(x H Cx) and (4.14) hods. Under the assumption (4.18), (4.40c) hods with α = 1. Thus g(1) = ς(x) 2 + 2ψ(x) + ϕ(x) > 0, or equivaenty the perturbed quadratic poynomia is sti hyperboic. Note (4.4) hods for ϕ(x) = 0. Thus δ 3 (x, ξ) 0 and δ 3 (x, ξ) 0. We can take, in (4.37), δ + ub (x) = δ+ 2 (x) = Cx xh, δ+ ς(x) ub (x) = δ 2 + (x) = xh Cx ς(x) (4.42) to give δ + ub (x) C 2 min x 0 ς 0 (x), δ + ub (x) C 2 min x 0 ς 0 (x). 29