Spectral sequences for quadratic pencils and the inverse spectral problem for the damped wave equation

J. Math. Pures Appl.,78, 1999,p. 965-980

SPECTRAL SEQUENCES FOR QUADRATIC PENCILSAND THE INVERSE SPECTRAL PROBLEM FOR THE

DAMPED WAVE EQUATION

Pedro FREITAS 1

Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa Codex, Portugal

Manuscript received 15 March 1999

ABSTRACT. – We give necessary conditions for a sequence of complex numbers closed under complexconjugation to be the spectrum of the weakly damped wave operator. These restrictions are a consequenceof some new results on spectral sequences for Hermitian quadratic pencils which are based on Weyl’s andMirsky’s classical eigenvalue inequalities. In the case of finite-dimensional weakly damped pencils ourconditions are both necessary and sufficient. We also obtain some conditions for overdamped pencils ofdegreem, and show that some of the inequalities that have to be satisfied in the weakly damped case arenow reversed. Elsevier, Paris

Keywords:Hermitian pencils, Damped wave equation, Weyl’s inequalities

RÉSUMÉ. – On établit des conditions nécessaires pour qu’une suite de nombres complexes ferméepour la conjugaison soit le spectre de l’opérateur des ondes faiblement amorties. Ces restrictions sont laconséquence de quelques nouveaux résultats pour des suites spectrales pour des polynômes hermitiensquadratiques, basés sur les inégalités classiques de Weyl et Mirsky pour les valeurs propres d’une matrice.Dans le cas de dimension finie, et quand le polynôme est faiblement amorti, nos conditions sont nécessaireset suffisantes. On obtient aussi des conditions pour des polynômes de degrém fortement amortis, et ondémontre que certaines inégalités qui sont verifiées dans le cas faiblement amorti sont maintenant inversées. Elsevier, Paris

Mots Clés:polynômes hermitiens, équation des ondes amorties, inégalités de Weyl

1. Introduction

LetH be a Hilbert space (finite- or infinite-dimensional) with inner product(· , ·) and considerthe quadratic pencil defined by:

L(λ)=Aλ2+Bλ+C,

whereA, B andC are self-adjoint operators defined on a dense subset ofH. The spectrumΣLof L is defined to be the subset of the complex plane formed by the complex numbersλ for whichL(λ) does not have a bounded inverse with dense domain inH. In the finite-dimensional case

1 Partially supported by FCT (Portugal) under projects PRAXIS/PCEX/P/MAT/36/96 and PRAXIS XXI Project2/2.1/Mat/199/94. E-mail: [email protected]

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966 P. FREITAS

or when the operatorsA, B andC satisfy some conditions, the spectrum will be discrete andconsist only of eigenvalues. In the former case, this corresponds to the roots of the characteristicpolynomialp(λ) := det[L(λ)].

Let nowΛ be a sequence of complex numbers which is closed under complex conjugation.We would like to consider the question of whether or not there exist self-adjoint operatorsA, BandC within certain classes such thatΣL = Λ. When this happens, we shall say thatΛ is anadmissible spectral sequence forL.

In the case whereA= 0 andB is the identity inH, for instance, the problem reduces to findingthe possible admissible spectral sequences for the self-adjoint operatorC. This is trivial in thegeneral finite-dimensional case (ann × n Hermitian matrix may have any sequence ofn realnumbers as eigenvalues), but if we require thatC belongs to special classes of matrices thenthe difficulty of the problem increases immediately. Two examples are, for instance, the inverseproblem for symmetric Toeplitz matrices [16] and certain classes of matrices arising in graphtheory [2].

The problem also becomes quite complicated in infinite-dimensional spaces. An important stepfor knowing which sequences are admissible in this case is to study the asymptotic behaviourof the spectrum. For the case of elliptic operators on bounded domains ofRn, for instance,there exists a wide range of literature dedicated to this problem—see [18] and the referencestherein. WhenH= L2(0,π) andC is taken to be of the form−∂2/∂x2+ q , with q in L∞(0,π)and homogeneous Dirichlet boundary conditions, for instance, a strictly increasing sequenceΛ= λj ∞j=1 of real numbers will be an admissible spectral sequence forL (for an appropriatefunctionq) if and only if

λj = 1

π

π∫0

q(s)ds + j2+ sj ,

as j goes to infinity, wheresj is in `2—see [21], for instance. We thus see that in this casethe asymptotic behaviour of the spectrum together with the fact that the spectrum is real and acondition on the multiplicities (all eigenvalues are simple) are the only restrictions on spectralsequences. The corresponding spectral problem in bounded domains inRm is still open, evenwhen only finite (truncated) spectral sequences are considered—see [1] for some related results.

In general, the problem of finding the possible admissible spectral sequences for a given classof quadratic pencils does not reduce to that of finding the admissible spectral sequences for aself-adjoint operator and little is known about the restrictions these sequences have to satisfy. Inparticular, complex eigenvalues may now exist and thus we should expect that extra (nontrivial)conditions on their relative placement should arise. This is of interest as, for instance, thereare several important problems such as stability and the rate of decay of solutions, which arerelated to global properties of the spectrum and depend on the supremum of the real parts of theeigenvalues [3–5,7,8,17].

In the finite-dimensional case ifA, B andC are allowed to be chosen independently, thenit is possible to have any sequence which is closed under complex conjugation as a spectralsequence forL. Here the problem should then be seen as that of determining the relationsbetween the eigenvalues of the quadratic polynomial and those of the matricesA, B andC.If, for instance,L is assumed to be monic (that isA is the identity matrix—ifA is positivedefinite this can be assumed without loss of generality), it is known that ifb1 (resp.c1) andbn(resp.cn) denote, respectively, the smallest and the largest eigenvalues ofB (resp. ofC), thenany nonreal eigenvalueλ of L has to satisfy:

−bn 6 2 Re(λ)6−b1 and c1/21 6 |λ|6 c1/2

n .(1)

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SPECTRAL SEQUENCES FOR QUADRATIC PENCILS 967

Similar bounds (except for that withcn) also hold, for instance, in the case of the wave equation.Here we takeH to be the spaceL2(Ω), Ω a bounded domain inRm with smooth boundary,Bthe multiplication operator by anL∞(Ω) functionb andC =−1D the Dirichlet Laplacian. Inthis case,b1 andbn are the essential infimum and supremum ofb, respectively, andc1 is theprincipal eigenvalue of the Laplacian. In the case whereΩ is a bounded interval on the real line,it is known that there are also certain asymptotic properties that a spectral sequence will have tosatisfy [3,8]. This example can be seen as the equivalent for quadratic pencils of operators of thelinear pencilIλ+1D + q mentioned above. From this we see that it is of interest to understandwhat the possible spectral sequences are when some of the operators (in this caseA andC) arefixed.

Typically, we should thus expect admissible spectral sequences for quadratic pencils to satisfytwo types of restrictions. On the one hand, there are restrictions placed on the position ofindividual eigenvalues such as those in (1). On the other hand, and in the infinite-dimensionalcase, there are restrictions on the possible asymptotic behaviour of such sequences. In the presentpaper we are concerned mainly the former type of restrictions in the weakly damped case.In particular, we shall prove that conditions (1) are a special case of a more general set ofmultiplicative and additive inequalities that have to be satisfied by spectral sequences. Theseinequalities turn to be closely related to Weyl’s and Mirsky’s classical inequalities between theeigenvalues of a general matrixM and those of the associated Hermitian matricesMM∗ and12(M +M∗)—see, respectively, [23,20]. In the self-adjoint finite-dimensional case and when allthe eigenvalues are nonreal, we show that these inequalities are also sufficient. It is an interestingopen question whether or not, together with the appropriate asymptotic restrictions, they will alsobe sufficient for certain situations in the infinite-dimensional setting such as the wave equation.

When there exist real eigenvalues the inequalities mentioned above will, in general, not besatisfied. It is possible, however, to show that when the pencil is overdamped (which, in particular,implies that the spectrum is now real and thus this cannot happen for a wave equation of theform above), then these inequalities are reversed. This raises the question of understanding whathappens in between these two situations when the spectrum consists of both real and nonrealeigenvalues. In the overdamped case, these inequalities are obtained as a corollary to a moregeneral result which holds for Hermitian pencils of degreem and which may be seen as ageneralization of the fact that, for a monic (scalar) polynomialp, the product of its roots isequal top(0).

The plan of the paper is as follows. We begin by establishing some notation and state the mainresults of the paper in Section 2. In Sections 3 and 4 we prove the main results for weakly dampedand overdamped pencils, respectively. In Section 5 we show that in the case of general quadraticpencils, apart from the fact that the number of zero eigenvalues ofLmust equal the dimension ofthe kernel of the matrixC or of a determinant condition, there are no other restrictions whenC isnot scalar. The proofs of the results for the weakly damped wave equation are given in Section 6,and finally, in Section 7, we discuss the results obtained and present some conjectures and openquestions related to inverse spectral problems for quadratic pencils.

2. Background and main results

Assume that dimH= n <∞ and thatA is ann× n positive definite Hermitian matrix. Underthese conditions, and for spectral purposes, the polynomialL may be assumed to be monic, thatis,

L(λ)= Iλ2+Bλ+C.

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968 P. FREITAS

Regarding the spectral properties ofL, there are two important situations which are related to thesign of the discriminant(Bu,u)2− 4(u,u)(Cu,u).

DEFINITION 2.1. –A quadratic pencilL(λ) of the form above is said to be weakly dampedif (Bu,u)2 < 4(u,u)(Cu,u) for every nonzerou. It is said to be overdamped if the reverseinequality is satisfied.

In what follows, we recall some basic properties of weakly damped and overdampedpolynomials which will be needed in the sequel. These results go back a long way, and a goodreference for our purpose is the book [9] which also includes a list of the original sources.

We begin by giving a useful characterization of weakly damped pencils.

PROPOSITION 2.1. –The polynomialL is weakly damped if and only if it does not have realeigenvalues. In particular, the Hermitian matrixL(λ) is positive definite for all realλ.

In order to state the results in the overdamped case, we shall consider the spectrum ofL

from a different point of view. Given a quadratic pencilL, let q(λ,σ ) := det(L(λ) − σI) andconsider the polynomial equationq(λ,σ ) = 0. For each real value ofλ this equation definesn(real) values (counting multiplicities) ofσ (the eigenvalues of the Hermitian matrixL(λ)) whichcan be ordered in such a way that the functionsσj :R→ R, j = 1, . . . , n, thus obtained areanalytic [14]. We shall call these functions the eigencurves of the pencilL. In this setting, thereal eigenvalues of a matrix polynomialL are just the zero set of the functionsσj , j = 1, . . . , n.

In some cases, we will also want to consider the eigencurves ofL in such a way that they areordered and will refer to them as the ordered eigencurves ofL. Now it is no longer possible toensure that these are analytic functions of the real variableλ, but they remain continuous. Bothconcepts can be extended in the obvious way to general self-adjoint polynomials.

THEOREM 2.2. –Assume thatL is overdamped. Then:(i) all eigenvalues ofL are real and negative;(ii) there exists a negative numberr such thatn eigenvaluesλ(1)1 , . . . , λ

(1)n are greater thanr

andn eigenvaluesλ(2)1 , . . . , λ(2)n are less thanr;

(iii) If σ(λ) is an analytic eigencurve ofA andσ(λ(i)j )= 0, thensgn[σ ′(λ(i)j )] = (−1)i+1.

For a proof of this result, as well as for a statement of condition (iii) in terms of the signcharacteristic of a matrix polynomial see [9]. In order to motivate the definition of overdampingfor a general self-adjoint pencil, we shall show that these conditions are also sufficient for aquadratic polynomial to be overdamped.

PROPOSITION 2.3. –Assume that conditions(i), (ii) and (iii) in Theorem2.2hold. ThenL isoverdamped.

Proof. –From (i) it follows thatC is positive definite, for otherwise there would exist realnonnegative eigenvalues. As the eigenvalues ofL(λ) can be separated in the way describedin (ii) and such that the derivatives of the eigencurves at the eigenvalues to the right ofr arenegative and those at the eigenvalues to the left ofr are positive, it follows thatL(r) is a negativedefinite matrix. This in turn implies thatB is positive definite, for otherwise there would exist avectoru such that(u,u)= 1, (Bu,u)6 0 and then(L(r)u,u)= r2+ r(Bu,u)+ (Cu,u) > 0.Finally assume that there existed a vectoru such that(u,u)= 1 and(Bu,u)26 4(Cu,u). Then(L(r)u,u)> 0, a contradiction. 2

This result together with Proposition 2.1 suggest the following definitions for weakly dampedand overdamped self-adjoint matrix polynomials of general degree.

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DEFINITION 2.2. –LetA(λ)= Iλm +Am−1λm−1+ · · · +A1λ+A0 where the matricesAj ,

j = 1, . . . ,m−1, are Hermitian. ThenA is said to be weakly damped if it has no real eigenvalues.It is said to be overdamped if:

(i) all its eigenvalues are real and negative;(ii) itsmn eigenvalues can be denoted byλ(i)1 , . . . , λ

(i)n , i = 1, . . . ,m, in such a way that there

exist numbersrj , j = 1, . . . ,m− 1, such that

λ(m)j < rm−1< λ

(m−1)j < rm−2< · · ·< r1< λ(1)j ,

for all j = 1, . . . , n, and(iii) if σk(λ

(i)j )= 0, thensgn[σ ′k(λ(i)j )] = (−1)i+1.

Remark2.1. – From this it follows that for overdamped pencils there existm+1 real numbersrm < rm−1 < · · · < r1 < r0 < 0 such that the matrixA(rj ) is positive definite ifj is even andnegative definite ifj is odd.

Remark2.2. – Note that this definition of weakly damped pencils still makes sense in the casewhereH is an infinite-dimensional Hilbert space.

Another concept that will be important is that of a principal subpencil. This is basically thepencil obtained by considering the principal submatrices of its coefficients corresponding to thesame set of indices.

DEFINITION 2.3. –LetA(λ)= Iλm +Am−1λm−1+ · · · +A1λ+A0 be a self-adjoint pencil

whereAj , j = 1, . . . ,m − 1, are n × n matrices. For every index setα ⊂ 1, . . . , n, denote

byA(α)j the principal submatrix ofAj that lies in the rows and columns indexed byα, and by

A(α)(λ) the principal subpencil:

A(α)(λ)= I (α)λm +A(α)m−1λm−1+ · · · +A(α)1 λ+A(α)0 .

Finally, we need to define two sequences of numbers associated with quadratic Hermitianpencils.

DEFINITION 2.4. –Denote the eigenvalues ofB andC in increasing order bybj andcj , j =1, . . . , n, respectively. Given a Hermitian pencilL(λ)= Iλ2+Bλ+C write its eigenvaluesλj ,j = 1, . . . ,2n, in such a way that|λ1|6 · · ·6 |λ2n|, and letp = (p1, . . . , p2n) be a permutationof (1, . . . ,2n) for whichRe(λp1)> · · ·>Re(λp2n). We define the numbers:

fk =k∏j=1

cj −k∏j=1

|λ2j−1λ2j |

and

gk =k∑j=1

[bj +Re(λp2j−1 + λp2j )

],

for k = 1, . . . , n.

We are now ready to state the main results of the paper.

THEOREM 1 (Weakly damped quadratic pencils). –Let B (resp.C) be a Hermitiann × nmatrix andΛ be a sequence of2n nonreal numbers closed under complex conjugation. Then


970 P. FREITAS

there exists a Hermitian matrixC (resp.B) such thatΛ is an admissible spectral sequencefor L(λ) = Iλ2 + Bλ + C if and only if the numbersgk (resp. fk) are nonpositive for allk = 1, . . . , n− 1 andgn (resp.fn) is zero.

From this result it is possible to obtain restrictions on the regions of the complex plane wherethe eigenvalues may lie.

COROLLARY 2. –Assume thatL is weakly damped and denote its eigenvalues byλ±j , j =1, . . . , n, such thatλ+j = λ−j and|λ±1 |6 · · ·6 |λ±n |. Then:

(k∏j=1

cj

)1/2k

6 |λ±k |6(

n∏j=k

cj

)1/2(n−k+1)

,

where thecj ’s are the eigenvalues ofC written in increasing order.Let nowp = (p1, . . . , pn) be a permutation of(1, . . . , n) for whichRe(λ±p1

)> · · ·>Re(λ±pn).Then

− 1

2(n− k + 1)

n∑j=k

bj 6Re(λ±pk

)6− 1

2k

k∑j=1

bj ,

where thebj ’s are the eigenvalues ofB written in increasing order.

For the case of overdamped pencils we have stronger necessary conditions which, moreover,can be extended to general self-adjoint pencils.

THEOREM 3 (Overdamped pencils). –Assume that then × n Hermitian pencilA(λ) =Iλm +Am−1λ

m−1+ · · ·+A1λ+A0 is overdamped. Denote its eigenvalues byλ(i)1 > · · ·> λ(i)n ,

i = 1, . . . ,m, in such a way that there exist numbersrj , j = 1, . . . ,m− 1, for which

λ(m)j < rm−1< λ

(m−1)j < rm−2< · · ·< r1< λ(1)j ,

for all j = 1, . . . , n. Then, denoting the eigenvalues ofA0 byσ16 σ26 · · ·6 σn, we have that:

k∏j=1

σj > (−1)mm∏i=1

(k∏j=1

λ(i)j

), k = 1, . . . , n,

with equality fork = n.Let now the eigenvalues ofAm−1 be denoted byγ16 · · ·6 γn. Then:

k∑j=1

γj >−m∑i=1

(k∑j=1

λ(i)j

), k = 1, . . . , n,

with equality fork = n.

As a direct consequence of this result, we have that the inequalities of Theorem 1 are reversedfor overdamped pencils.

COROLLARY 4. –For an overdamped quadratic pencilfk andgk are nonnegative for allk,and zero fork = n.

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Finally, we have that part of the results in Theorem 1 can be extended to the weakly dampedwave equation.

THEOREM 5 (Weakly damped wave equation). –Consider the quadratic pencil:

L(λ)= Iλ2+ 2bλ−1Don a bounded domain inRn. Assume thatb is such thatL is weakly damped and denote its

eigenvalues byλ±j such that|λ±1 |6 |λ±2 |6 · · · andλ+j = λ−j . Then we have that the numbersfkare nonpositive for allk in N.

This result still holds if instead of−1D we consider more general classes of (positive) ellipticself-adjoint operators.

As in the finite-dimensional case, these inequalities give restrictions on the regions of thecomplex plane where the eigenvalues may lie. The following result is a direct consequence ofTheorem 5, the proof being similar to that of the multiplicative part of Corollary 2.

COROLLARY 6. –LetΩ in the previous theorem be the interval(0,π). Then:∣∣λ±k ∣∣> (k!)1/k.3. Weakly damped pencils

In the case of weakly damped pencils there are no restrictions on the definitness of thecoefficients of the polynomial, except in the case ofA0 which must be positive definite forotherwise there would exist real eigenvalues.

We begin by noting that every principal subpencil of a weakly damped pencil is also weaklydamped. This result will be used in Section 6 in the proof of Theorem 5.

LEMMA 3.1. –If L is ann× n weakly damped pencil, then anyk × k principal subpencil isalso weakly damped.

Proof. –This follows directly from the fact that the eigencurves of principal subpencils arebetween the eigencurves ofL, which in turn is a consequence of interlacing inequalities betweeneigenvalues of Hermitian bordered matrices (see, for instance, [12]). Since the eigencurves ofL

are above the horizontal axis, the same follows for those of the principal subpencil.23.1. Multiplicative inequalities

In this section we prove the multiplicative part of Theorem 1, that is, we show that given amatrix C and a sequenceΛ satisfying the hypothesis in that theorem, there exists a matrixB

such thatΛ is the spectrum of the pencilL(λ)= Iλ2+Bλ+C if and only if all the numbersfkare nonpositive andfn is zero. The proof follows an idea already used in [7]. The fact that thenumbersfk are all nonpositive follows from Weyl’s inequalities between the eigenvalues and thesingular values of a matrix. For the converse, we use a lemma due to A. Horn that states thatWeyl’s inequalities are also sufficient—see [11,13].

Assume that the sequences(c1, . . . , cn) and(λ1,Sλ1, . . . , λn,Sλn) are such thatfk 6 0 for all k,with equality fork = n. Then, by Horn’s lemma, there exists an upper triangular matrixJ whichhasc1/2

1 , . . . , c1/2n as its singular values andλ1, . . . , λn as its eigenvalues. LetP = J ∗J . ThenP

hasc1, . . . , cn as its eigenvalues and thus there exists a unitary matrixR such thatC =R∗PR.


972 P. FREITAS

Let nowW = R∗JR. ThenW andW∗ haveλ1, . . . , λn andSλ1, . . . ,Sλn as their eigenvalues,respectively. This gives that the polynomial(Iλ−W∗)(Iλ−W)= Iλ2− (W∗ +W)λ+W∗Whas the desired spectrum. On the other hand,W∗W = R∗J ∗RR∗JR = C and thus takingB =−(W∗ +W) gives the desired result.

Assume now that the spectrum of the Hermitian quadratic pencilL is given by the sequenceΛ= (λ1, . . .λn) and its conjugate, and that this sequence is ordered such that|λ1|6 · · ·6 |λn|.As L is weakly damped, it can be factorized asL(λ)= Iλ2+ Bλ+ C = (Iλ−W∗)(Iλ−W),whereW has the sequenceΛ as its spectrum. In other words, there exists a matrixW such that

B =−(W +W∗) and C =W∗W.

From this factorization, it follows that the spectrum ofL is the union of the spectra ofWandW∗, and that the singular values ofW are the square roots of the eigenvalues ofC. Hence,by Weyl’s inequalities between the eigenvalues and singular values of a matrix it follows that thenumbersfk are all nonpositive. The equality fork = n follows directly from the equality for thedeterminants. This proves the multiplicative part of Theorem 1.

We shall now prove the multiplicative part of Corollary 2—the additive part can be proven ina similar fashion.

Proof of Corollary 2. –From the fact that the numbersfk ’s are nonpositive fork = 1, . . . , n−1, andfn = 0, we have that:

k∏j=1

∣∣λ±j ∣∣2> k∏j=1

cj , k = 1, . . . , n− 1, andn∏j=1

∣∣λ±j ∣∣2= n∏j=1

cj .(2)

As |λ±j |6 |λ±j+1|, the first inequality follows.For the second inequality, notice first thatC is positive definite and soL has no zero

eigenvalues. Then, from (2) it follows that:

∣∣λ±k ∣∣2(n−k+1) 6n∏j=k

∣∣λ±j ∣∣26 n∏j=k

cj , k = 1, . . . , n,

which gives the desired result.23.2. Additive inequalities

We noe assume thatB and a sequenceΛ satisfying the conditions in Theorem 1 are given.The proof of the fact that it is a necessary and sufficient condition forΛ to be the spectrumof the corresponding quadratic pencil that the numbersgk are nonpositive andgn is zero isbased on Mirsky’s inequalities between the eigenvalues of ann× n matrixX and those of theHermitian matrixH = 1

2(X+X∗) [20]. Assume thatX has eigenvaluesx1, . . . , xn, ordered suchthat Re(x1)6 · · ·6 Re(xn), and denote the eigenvalues ofH by h16 · · ·6 hn. Then, Mirsky’sresult states that:

k∑j=1

Re(xj )>k∑j=1

hj , k = 1, . . . , n,

with equality fork = n. Conversely, if the sequenceshj andxj satisfy the inequalities above,then there exists a matrixX such thatX andH have these sequences as their eigenvalues.

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As in Section 3.1, we writeL(λ) = Iλ2 + λB + C as (Iλ −W∗)(I − λW), from which itfollows thatB = −(W +W∗) andC = W∗W . If the eigenvalues ofW arew1, . . . ,wn withRe(w1)6 · · ·6Re(wn) Mirsky’s result gives that:

−2k∑j=1

Re(wj )>k∑j=1

bj , k = 1, . . . , n,

with equality fork = n. Since for this pencil the eigenvalues arew1, . . . ,wn, Sw1, . . . , Swn, thisgives that the numbersgk , k = 1, . . . , n− 1, are nonpositive and thatgn is zero.

Assume now that we are given a matrixB with eigenvaluesb1 6 · · · 6 bn and a sequenceof nonreal eigenvaluesΛ = (λ1, . . . , λ2n) closed under complex conjugation and satisfyingRe(λ1)6 · · ·6Re(λ2n). Then it is always possible to define a sequenceΓ = (w1, . . . ,wn) suchthat Re(w1) 6 · · ·6 Re(wn) andΛ= (Γ, SΓ ), whereSΓ denotes the sequence obtained fromΓby taking complex conjugates. Then, if numbersgk corresponding to the eigenvalues ofB andthe sequenceΛ are nonpositive, using the converse of Mirsky’s result, we obtain that there existsa matrixW with Γ as its spectrum and such thatB =−(W∗ +W). Thus, it is enough to takeC =W∗W to obtain a Hermitian quadratic pencil with the desired spectrum. This concludes theproof of Theorem 1.

4. Overdamped pencils

We begin by showing that all the matrix coefficients of a general Hermitian overdamped pencilmust be positive definite. This can be seen as a generalization of the fact that a monic polynomialwhose zeros are all real and negative must have positive coefficients—see [19], for instance.

PROPOSITION 4.1. –LetA(λ)= Iλm +Am−1λm−1+ · · · +A1λ+A0 be overdamped. Then

Aj is positive definite for allj = 1, . . . ,m− 1.

Proof. –Let u be any unit vector and build the polynomial

a(λ)= λm + am−1λm−1+ · · · + a1λ+ a0,

whereaj = (Aju,u), j = 1, . . . ,m− 1. For each real numbersλ0, a(λ0) must lie between thelargest and the smallest eigenvalues ofA(λ0). Thus,

minjσj (λ)6 a(λ)6max

jσj (λ),

for all realλ. From Remark 2.1 it now follows thata must havem real negative zeros. This in turnimplies that the coefficientsaj , j = 1, . . . ,m−1, must all be positive and the result follows.2

A crucial fact in the proof of Theorem 3 is that every principal subpencil of an overdampedpencil is also overdamped.

LEMMA 4.2. –If L is ann× n overdamped pencil, then anyk× k principal subpencil is alsooverdamped.

Proof. –It is enough to show that ifL(α) is an (n − 1)× (n − 1) principal subpencil ofL,then it must be overdamped. Denote the eigencurves ofL andL(α), ordered in increasing order,by γj (λ), j = 1, . . . , n, andγ (α)j (λ), j = 1, . . . , n− 1, respectively. Then, from the interlacing


974 P. FREITAS

inequalities between eigenvalues of (Hermitian) bordered matrices, for every real numberλ wehave that:

γ1(λ)6 γ (α)1 (λ)6 γ2(λ)6 · · ·6 γn−1(λ)6 γ (α)n−1(λ)6 γn(λ).This, together with the fact thatL is overdamped, implies that there are(n− 1)m zeros of theeigencurvesγ (α)j , j = 1, . . . , n− 1, corresponding to the eigenvalues ofL(α). Clearly these areall real and negative.

We have that the eigenvalues ofL can be denoted byλ(k)j , j = 1, . . . , n, k = 1, . . . ,m, insuch a way that they satisfy the condition (ii) in the definition of overdamped pencils. Sinceany ordered eigencurve ofL(α) lies between two of the ordered eigencurves ofL(α), it followsthat between two consecutive eigenvalues ofL with the same upper indexk, there always liesone eigenvalue ofL(α)—note that ifL has an eigenvalueλ0 with multiplicity µ, thenL(α) willhave an eigenvalue with multiplicity at leastµ− 1 and at mostµ+ 1. This shows that any set ofnumbersrj , j = 1, . . . ,m−1, that can be used to separate the eigenvalues ofL will also separatethose ofL(α).

Finally, note that if any of the derivatives of the (analytic) eigencurves ofL(α) were to be zeroor to have the wrong sign at an eigenvalue ofL(α), this would imply the existence of more than(n− 1)m eigenvalues for this principal subpencil.24.1. Multiplicative inequalities

The fact that any principal subpencil of an overdamped pencil is also overdamped, togetherwith the interlacing inequalities between the eigenvalues of a pencil and a subpencil used inthe proof of Lemma 4.2, allow us to establish the set of inequalities for the eigenvalues ofoverdamped pencils stated in Theorem 3.

Proof of the multiplicative part of Theorem 3. –The proof is by induction inn. Clearly theresult is true whenn is one, as this is just saying that the product of the roots of a polynomialp

is equal top(0). Assume now that the result is true forn − 1. Without loss of generality,we may assume that a change of basis has been carried out such thatA0 = diagσ1, . . . , σn,with 0< σ1 6 · · ·6 σn. Consider the principal subpencilA(α) with α = (1, . . . , n− 1), that is,the pencil obtained fromA by deleting the last row and column of each of the matricesAj .From Lemma 4.2, we have thatA(α) is overdamped. Hence its eigenvalues may be denotedby µ(i)1 > · · · > µ(i)n−1, i = 1, . . . ,m, and, from the proof of that lemma, we also have that thenumbersrj that separate the eigenvalues ofA also separate those ofA(α):

µ(µ)j < rm−1<µ

(m−1)j < rm−2< · · ·< r1<µ(1)j ,

for j = 1, . . . , n− 1.From the interlacing properties of eigencurves of subpencils, we have that:

−µ(k)j >−λ(k)j (> 0),(3)

for all j = 1, . . . , n− 1 andk = 1, . . . ,m.From the fact that the result holds forn− 1 and that the eigenvalues ofA(α)0 areσ1, . . . , σn−1,

it follows thatk∏j=1

σj > (−1)mm∏i=1

(k∏j=1

µ(i)j

), k = 1, . . . , n− 1,

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with equality whenk = n− 1. Using now (3) we have that

(−1)mk∏j=1

µ(i)j > (−1)m

k∏j=1

λ(i)j , i = 1, . . . , n− 1.

Combining these last two inequalities gives the desired result fork = 1, . . . , n−1. Equality in thecase wherek = n follows directly from the fact that the product of the roots of the characteristicpolynomial ofA is equal to the determinant ofA0. 24.2. Additive inequalities

The proof of the additive inequalities in Theorem 3 is similar to that of its multiplicativecounterpart. The result clearly holds whenn= 1, as this is just saying that the sum of the roots ofa polynomial of degreem and with leading coefficient 1 is minus the coefficient of the(m− 1)thpower. Assume now that the result holds forn− 1. Without loss of generality, we may assumethatAm−1 = diagγ1, . . . , γn with 0< γ16 · · ·6 γn. As in the proof of the multiplicative partof the theorem, we now consider the principal subpencilA(α) with α = (1, . . . ,m− 1), which,by Lemma 4.2, is also overdamped. Denote its eigenvalues byµ

(i)1 > · · ·> µ(i)n−1, i = 1, . . . ,m.

As before, we have that (3) holds. Since the result is true forn− 1, we also have that:

k∑j=1

γj >−m∑i=1

(k∑j=1

µ(i)j

), k = 1, . . . , n− 1,

with equality fork = n− 1. From this and (3) the result follows fork = 1, . . . , n− 1. That theequality holds whenk = n follows from the fact that the sum of all the eigenvalues ofL equals− trace(Am−1).

5. General quadratic pencils

In this section we consider the general case of a quadratic polynomialL(λ)= Iλ2+Bλ+Cand show that if it is not assumed thatL is Hermitian, then there are no restrictions on the possiblespectral sequences whenC is given, apart from the fact that if this matrix is singular then therehave to exist zero eigenvalues. The proof of this result is based on the fact that it is possible tofactorize any given (nonscalar) nonsingularn×nmatrixC into twon×nmatrices with arbitraryeigenvalues, as long as their product is equal to the determinant ofC.

THEOREM 5.1 (Sourour). –Let C ∈Mn be a given matrix and supposerank(C) = k 6 n; ifk = n, assume thatC is not a scalar matrix. Letx1, . . . , xn and y1, . . . , yn be given complexnumbers, exactlyn− k of which are zero; if k = n assume that:

n∏j=1

xjyj = detC.

Then there exist matricesX,Y ∈Mn with eigenvaluesx1, . . . , xn and y1, . . . , yn, respectively,such thatC =XY .

For a proof see [13,22]. With this result, we are able to prove the following:


976 P. FREITAS

PROPOSITION 5.2. –LetL(λ)= Iλ2+λB+C be a quadratic polynomial and assume thatC

is given. Then, ifC is not scalar andrank(C)= k, for any sequenceΛ of 2n complex numbersexactlyn− k of which are zero there exists a matrixB ∈Mn such thatL hasΛ as its spectrum.

Let nowC = cI (c 6= 0). Then, ifΛ can be written as(x1, . . . , xn, y1, . . . , yn) in such away thatxjyj = c for all j = 1, . . . , n, the same result holds andB can now be chosen tobeB = diag−(x1+ y1), . . . ,−(xn + yn).

Proof. –Given a matrixC and a sequenceΛ = (x1, . . . , xn, y1, . . . , yn) which satisfy theconditions in the proposition, letX andY be matrices as in Theorem 5.1. Then the polynomial(Iλ− x)(Iλ−Y )= Iλ2− (X+Y )λ+XY has the desired spectrum. SinceXY = C, this showsthat takingB =−(X+ Y ) ensures that the spectrum ofL isΛ.

WhenC = cI , we have thatL(λ)= I (λ2+ c)+Bλ. From this it follows that the eigenvaluesof L must satisfy the quadratic equationλ2 + biλ + c = 0, where thebi ’s are the eigenvaluesof B. Clearly ifB is as in the hypothesis the eigenvalues ofL are as desired.2

From the point of view of completion of partially specified matrices, this result states thatgiven a matrixC and a matrix of the form

M = 0 I

−C −B

it is possible to prescribe any spectrum satisfying the conditions in the proposition above bychoosing the matrixB in an appropriate fashion.

6. The wave equation

We now extend some of the results for weakly damped quadratic pencils of matrices to thecase of the wave equation. In order to do this, we shall use a discretization of the eigenvalueproblem associated with the wave equation based on Fourier series and then use the results inSection 3.1.

Consider the eigenvalue problem:

L(λ)u= λ2u+ 2λb(x)u−1Du= 1, x ∈Ω,(4)

whereH is taken to beL2(Ω). Denote byγk∞k=1 the eigenvalues of the Dirichlet Laplacianin Ω and byψk∞k=1 the (complete orthonormal) set of corresponding eigenfunctions. We nowconsider the following truncation of ordern

Ln(λ)Un = (Inλ2+Bnλ+Cn)Un = 0,(5)

whereBn = (bij )ni,j=1 with bij = (bψi,ψj ), andCn = diagγ1, . . . , γn. In the one-dimensionalcase which, with the exception of some special geometries, is the only instance where one canactually obtain explicit expressions for these matrices, we have (on(0,π)) ψk = sin(kx) andγk = k2. By writting b as a Fourier cosine series, that is:

b(x)= b0

2+∞∑k=1

bk cos(kx),

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we obtain

Bn =

b0− b2 b1− b3 b2− b4 · · · bn−1− bn+1

b1− b3 b0− b4 b1− b5 · · · bn−2− bn+2

b2− b4 b1− b5 b0− b6 · · · bn−3− bn+3

...

bn−1− bn+1 bn−2− bn+2 bn−3− bn+3 · · · b0− b2n

(6)

andCn = diag1,4, . . . , n2.We begin by showing that if the original problem has only nonreal eigenvalues, then the same

is true of (5).

PROPOSITION 6.1. –If the quadratic pencilL in (4) is weakly damped, then the truncatedpencilLn is also weakly damped.

Proof. –We show that the result holds ifn is large enough. Then, by Lemma 3.1 it follows thatit holds for alln, as the truncation is built in such a way thatLn−1 is a principal subpencil ofLn.

Any real eigenvalue of the discretized problem(Inλ

2+Bnλ+Cn)φ = 0

may be obtained as a solution of the equationσ(p)= p2, whereσ denotes the eigenvalues of theone-parameter eigenvalue problem

(−Cn + pBn)φ = σ(p)φandλ=−p. Note that this corresponds to the discretization of the problem:

1Du+ pb(x)u= σ(p)u.(7)

Denoting byσj andσj , respectively, the eigencurves of the infinite-dimensional problem (7) andthose of the corresponding discretized problem, it is thus sufficient to show that if the eigencurvesσj do not intersect the parabolaσ = p2, then for large enoughn the eigencurvesσj (p) alsoremain below the same parabola.

Whenp = 0, we haveσj = σj = −γj , independently of the order of the truncation. On theother hand, and since the operator−Cn + pBn is self-adjoint for realp, we have that the curvesσj may be chosen to be analytic. A simple calculation gives that [6].

σ ′(p)= φ∗Bnφ,where it is assumed thatφ∗φ = 1. SinceBn is the restriction of a bounded operator to a finite-dimensional subspace ofH we have that there exists a constantM, independent ofn, such that|σ ′(p)|<M. Let n1 be such that the straight lines starting at−γn1 and with slope±M do notintersect the parabolaσ = p2. Then the eigencurves starting at eigenvalues for whichσj (0) isless than or equal to−γn1 remain below the parabola.

Since the growth of the eigencurves of the original problem is slower thanp2 [4], we havethat there existsp∗ such that, for allj less thann1, the straight lines passing through the pointσj (±p∗) with slope±M (respectively) do not intersect the parabola.


978 P. FREITAS

Given a positive numbersδ, let nown= n(δ) be such that the eigencurves of the discretizedproblem up ton1 areδ-close to the corresponding eigencurves of the original problem on theinterval(−p∗,p∗), that is:∣∣σj (p)− σj (p)∣∣6 δ, p ∈ (−p∗,p∗), j = 1, . . . , n1.

That this is always possible, follows from results in the theory of approximation of eigenvalues ofself-adjoint operators via spectral methods—see, for instance, [15]. This implies that by choosingδ sufficiently small (and thusn large enough) if there are no intersections of the eigencurvesσjwith the parabola, the same is true for the eigencurvesσj . 2

Proof of Theorem 5. –Assume that the pencilL defined in (4) is weakly damped. ByProposition 6.1 this implies that the corresponding pencilLn is also weakly damped. Denotethe eigenvalues ofL as in the hypothesis of Theorem 5, and those ofLn by λ±j such that

|λ±1 |6 |λ±2 |6 · · ·6 |λ±n |.Assume now that there existed an integerN for whichfN was strictly positive, that is

N∏j=1

cj >

N∏j=1

∣∣λ+j ∣∣2.From results in [15] we have that given any positive numberδ there existsn large enoughsuch that each ball of radiusδ centered at the eigenvaluesλ±j , j = 1, . . . ,N , will contain oneeigenvalue of the pencilLn. There are now two cases to consider. Either these 2N eigenvaluesare the first 2N eigenvalues of the pencilLn or not. In the first case, if follows directly that bychoosingδ small enough one would have to have thatfN would also be positive for the pencilLn,which is impossible by Theorem 1. In the second case, which corresponds to the existence ofspurious eigenvalues in the approximation pencil, we have eigenvalues which are not necessarilyclose to anyλ±j . However, if they are outside any of the balls of radiusδ they will have a smaller

absolute value than the eigenvalue with the same index, that is,|λ±j |> |λ±j |. Thus, if one of thesespurious eigenvalues occurs, then a similar inequality will also hold for eigenvalues with largerindexj and we have that

N∏j=1

cj >

N∏j=1

∣∣λ+j ∣∣2> N∏j=1

∣∣λ+j ∣∣2.Again, this would contradict Theorem 1.2

7. Discussion and open problems

The results obtained stress the fact that the structure of the spectrum of a quadratic polynomialis quite different depending on whether it is weakly damped or overdamped. Furthermore, theyraise the question of how the transition between the two situations is done and what restrictionsthe spectrum has to satisfy when, for instance, there exist both real and nonreal eigenvalues. Onepossible way of addressing this question is by considering polynomials depending on a parameterand then try to follow the changes in the spectrum as this parameter causes the polynomialto change from weakly damped to overdamped. To this end, consider a Hermitian quadraticpolynomial of the formLx(λ)= Iλ2+ xBλ+ C, wherex is a positive parameter andB andCare positive definite. By increasing the parameterx from zero we see thatLx changes from being

TOME 78 – 1999 –N 9


weakly damped to being overdamped whenx is large enough. We then know that, for instance,the numbersfk must go from being nonpositive for smallx to being nonnegative for largex.

This type of pencils was studied in [10] and, in particular, it was shown through examples thatthe number of real eigenvalues is not necessarilly an increasing function ofx. It follows directlyfrom Theorem 3.3 in [6] that if there are real eigenvalues for a certain value of the parameter,there will always exist real eigenvalues for all larger values ofx. It is also not difficult to seethat (in the quadratic case) once a pencil is overdamped for a certain value ofx, it will remainoverdamped for all larger values of this parameter. Thus a first remark is that for every quadraticHermitian pencilLx there exist positive real numbersx0 andx1 such thatLx is weakly dampedif and only if x is in [0, x0) and overdamped if and only ifx is in (x1,+∞). Note that havingonly real eigenvalues is not a sufficient condition for a pencil to be overdamped. The followingexample shows that having real negative spectrum is not a sufficient condition for the numbersfk to be nonnegative either.

Consider the 2× 2 quadratic polynomialL(λ)= Iλ2+ xBλ+C, where:

B =7 .3

.3 13

and C =12 0

0 43

.In this case, forx ∈ (1,1.02) all eigenvalues are real but the system is not overdamped.Furthermore, in this range we have thatf1 = c1 − λ1λ2 < 0, which shows that having onlyreal eigenvalues is not enough for the inequalities in Theorem 3 to hold.

Another set of questions is related to whether or not the restrictions obtained are also sufficient.Here we have shown that is the case for weakly damped Hermitian pencils in a finite-dimensionalsetting. In many applications, it is important that the matrices involved are actually real. Weconjecture that the result still holds in this case, that is, that Theorem 1 still holds if insteadof Hermitian we consider real symmetric matrices. We remark that in the very special caseconsidered in [7], where the spectrum reduces to a point on the real axis (and thus the pencil isneither weakly damped nor overdamped) it is shown that this may be achieved via real symmetricmatrices. It is not difficult to see that the results obtained here can be extended to situationssimilar to this one. Regarding overdamped pencils, it would be of interest to know whether ornot the more restrictive conditions given by Theorem 3 are also sufficient.

Once we step into the infinite-dimensional setting, the whole problem becomes much morecomplicated. In particular it does not seem likely that, even in the one-dimensional case, theconditions given in Theorem 5 (together with the asymptotic results from [3,8]) will also besufficient. This is due to the fact that, as can be seen from the expression forBn in (6), themultiplication operators considered are linear operators of a very special form. In particular, thisalso means that one cannot expect estimates such as that in Corollary 6 to be very good when itcomes to high-frequency eigenvalues. That this is the case may be confirmed by looking at themuch more precise asymptotic estimates available [3,8].

Again, it would be quite interesting to understand what may happen when there exist both realand nonreal eigenvalues.

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[2] Y. COLIN DE VERDIÈRE, Multiplicities of eigenvalues and tree-width of graphs,J. Comb. Theory B74 (1998) 121–146.


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[3] S. COX and E. ZUAZUA, The rate at which energy decays in a damped string,Comm. PartialDifferential Equations19 (1994) 213–243.

[4] P. FREITAS, On some eigenvalue problems related to the wave equation with indefinite damping,J. Differential Equations127 (1996) 320–335.

[5] P. FREITAS, Optimizing the rate of decay of solutions of the wave equation using genetic algorithms:A counterexample to the constant damping conjecture,SIAM J. Control Optim.39 (1999) 376–387.

[6] P. FREITAS, M. GRINFELD and P.A. KNIGHT, Stability for finite-dimensional systems with indefinitedamping,Adv. Math. Sci. Appl.7 (1997) 435–446.

[7] P. FREITAS and P. LANCASTER, On the optimal value of the spectral abscissa for a system of linearoscillators,SIAM J. Matrix Anal. Appl., to appear.

[8] P. FREITAS and E. ZUAZUA, Stability results for the wave equation with indefinite damping,J.Differential Equations132 (1996) 338–352.

[9] I. GOHBERG, P. LANCASTER and L. RODMAN, Matrix Polynomials, Academic Press, New York,1982.

[10] I. GOHBERG, P. LANCASTERand L. RODMAN, Quadratic matrix polynomials with a parameter,Adv.Appl. Math.7 (1986) 253–281.

[11] A. HORN, On the eigenvalues of a matrix with prescribed singular values,Proc. Amer. Math. Soc.5(1954) 4–7.

[12] A. HORN and C.A. JOHNSON, Matrix Analysis, Cambridge Univ. Press, Cambridge, 1985.[13] A. HORN and C.A. JOHNSON, Topics in Matrix Analysis, Cambridge Univ. Press, Cambridge, 1995.[14] T. KATO, Perturbation Theory of Linear Operators, Springer, New York, 1980.[15] M.A. K RASNOSEL’ SKII , G.M. VAINIKKO , P.P. ZABREIKO, YA.B. RUTITSKII and V.YA. STET-

SENKO, Approximate Solution of Operator Equations, Wolters-Noordhoff Publishing, Groningen,1972.

[16] H.J. LANDAU, The inverse eigenvalue problem for real symmetric Toplitz matrices,J. Amer. Math.Soc.7 (1994) 749–767.

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Documents

Spectral sequences for quadratic pencils and the inverse spectral problem for the damped wave equation