doi:10.1017/S144618110800031X
PERTURBATION RESULTS RELATED TO PALINDROMIC
EIGENVALUE PROBLEMS
E. K.-W. CHU˛ 1, W.-W. LIN2and C.-S. WANG3
(Received 13 September, 2007; revised 30 September, 2008) Abstract
We investigate the perturbation of the palindromic eigenvalue problem for the matrix quadratic P(λ) = λ2A?
1+λA0+A1with A0, A1∈C
n×n and A?
0=A0(where? = T or H ). The perturbation of eigenvalues in the context of general matrix polynomials, palindromic pencils, (semi-Schur) anti-triangular canonical forms and differentiation is discussed.
2000 Mathematics subject classification: primary 15A18, 15A22, 65F15.
Keywords and phrases: anti-triangular form, eigenvalue, eigenvector, matrix polynomial, palindromic eigenvalue problem, palindromic linearization, palindromic pencil, perturbation.
1. Introduction Consider the matrix quadratic
P(λ) ≡ λ2A?1+λA0+ A1,
where A0, A1∈Cn×nwith A?0=A0(? = T or H), and the corresponding palindromic
quadratic eigenvalue problem
P(λ)x = 0, x 6= 0. (1.1)
In this paper we consider only regular matrix polynomials P(λ), where “regular” is understood to mean det P(λ) 6≡ 0.
1School of Mathematical Sciences, Building 28, Monash University, VIC 3800, Australia;
e-mail:[email protected].
2Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan;
e-mail:[email protected].
3Department of Mathematics, National Cheng Kung University, Tainan 701, Taiwan;
e-mail:[email protected]. c
Australian Mathematical Society 2009, Serial-fee code 1446-1811/09 $16.00
From the transpose or Hermitian of (1.1), a palindromic eigenvalue problem is seen to possess a spectrum σ (P) that contains both λ and its “reciprocal” 1/λ? (with 0 and ∞ considered to be reciprocal to each other). Under favourable conditions, the eigenvalue problem of the original matrix polynomial P(λ) has a palindromic linearization of the formλZ ± Z?[6,10,16].
We can transformλZ − Z?to the formν(−Z) + (−Z)?, withν = −λ. Similarly, λ2A?
1+λA0+A1 andν2A?1−ν A0+A1 define equivalent palindromic eigenvalue
problems. As a consequence, and for simplicity of presentation, we shall concentrate on the palindromic eigenvalue problem. Anti-palindromic or odd/even eigenvalue problems [16–18] can be treated similarly.
A solid foundation for the solution of palindromic eigenvalue problems has been laid by Hilliges, D.S. Mackey, N. Mackey, Mehl and Mehrmann (see [13,16, 17]). An alternative approach to tackling the problem which involves structure-preserving doubling algorithms can be found in [6]. Recently, quadratic eigenvalue problems [21] have attracted much interest. An important example of palindromic eigenvalue problems can be found in the vibration analysis of fast trains; see [13] or [14] for a general introduction and [12] for details. For the general perturbation of eigenvalues for polynomial eigenvalue problems, see [5,20]; consult also [8,15] for some related results.
This paper is organized as follows. In Sections 2–5, the perturbation of eigenvalues in terms of general matrix polynomials, palindromic linearizations, the anti-triangular canonical form [16–18] and the semi-Schur anti-triangular canonical form is investigated by means of the Bauer–Fike technique for perturbations of arbitrary size [5]. The derivatives of eigenvalues and eigenvectors of P(λ) are considered in Section6. We summarize our conclusions in Section7. The results in Section3–5are applicable to the more general palindromic pencils Z −λZ?, which may not be linearizations of any matrix polynomials.
Note that Sun and Stewart’s implicit function approach [19] has been applied to palindromic linearizations, general matrix quadratics and palindromic eigenvalue problems in [7], to obtain perturbation results for (simple) eigenvalues and the corresponding deflating subspaces.
It is important to distinguish between the different perturbation techniques. Note that the Bauer–Fike technique allows perturbations of arbitrary size, and the clustering of eigenvalues (and hence the corresponding deflating subspaces) may vary greatly. Consequently, it is meaningless to talk about perturbation of eigenvectors or deflating subspaces in the Bauer–Fike-type perturbation theorems of Sections2–5. Perturbation results obtained via differentiation or implicit function approaches are valid only for asymptotically small perturbations, but results for deflating subspaces are available. Note that only simple eigenvalues (or the sums and averages of multiple eigenvalues) are differentiable, so generalized derivatives (or subgradients) have to be utilized in general [4]. We remark that the naive differentiation technique employed in Section6
assumes differentiability, which can only be proved rigorously by using tools such as the implicit function theorem.
Next, we give a few words of warning. Comparison of perturbation results is a risky art. Typically, error bounds and condition numbers are simplified upper bounds of more complicated quantities, and a better (worse) upper bound does not always imply a smaller (bigger) error. Furthermore, optimization of such upper bounds, though often possible, is seldom attempted because of cost or inconvenience, making comparisons of perturbation results even more perilous. Therefore, we do not claim to have found the “best” perturbation results, if such exist at all. We shall quite often interpret perturbation results qualitatively, rather than apply them quantitatively, and we will indicate when things may go wrong or pitfalls to avoid. Nevertheless, our perturbation results, in addition to those in [2, 7, 15], are among the very few that are currently available for palindromic eigenvalue problems, and should be of use in related investigations. Lastly, conditions qualifying when perturbations are large or (asymptotically) small can be written down but are complicated and rarely checked. Again, such perturbation results may have to be used qualitatively rather than quantitatively.
2. Bauer–Fike theorem for general matrix polynomials
The unstructured perturbation result for general matrix polynomials, presented in Theorem 2.1 below, may not be directly applicable or satisfactory for palindromic eigenvalue problems. However, it serves as a reference for the structured perturbation results that we consider in later sections. Also, palindromic eigenvalue problems are sometimes perturbed in an unstructured manner; one example is when the QZ algorithm [11] is applied to an associated palindromic linearization Z −λZ?. The associated perturbation problem has to be treated as an unstructured one, using the theorem below.
We now state, without proof, [5, Theorem 4.2] on the perturbation of eigenvalues of a general matrix polynomial.
THEOREM2.1. Consider a regular matrix polynomial L(α, β) ≡
l
X
j =0
Bjαjβl− j
and its perturbation
eL(α, β) ≡ l X j =0 e Bjαjβl− j, Bej ≡Bj +δBj ( j = 0, . . . , l).
Let(X, T, Z) be a resolvent triple for L (see [5,10]) which is constructed using some finite and infinite Jordan pairs, JF and J∞. For (αi, βi) ∈ σ(L) and (α, β)
∈σ (eL) with the scaling |αi|2+ |βi|2=1 = |α|2+ |β|2, the spectral variation of eL
from L is defined as
sL(eL) ≡ max
(α,β){s(α,β)}, s(α,β)≡mini
Let p be the maximum dimension of the Jordan blocks in JF or J∞.
Then, for k · k = k · kτ (whereτ = 1, 2 or ∞),
s(α,β)≤maxnθ1, θ11/po, θ1≡pFκ1, (2.1) whereF≡c1 √ (l + 1)/2 with c1=1 (forτ = 1, 2) or c1= √ l (forτ = ∞) and κ ≡ kXk · kZk, 1 ≡ k[δB0, . . . , δBl]k. Also, sL(eL) ≤ maxnθ1, θ11/po. (2.2) COMMENTS2.2.
(a) Note that, in the above theorem, we use the representation(α, β) for λ = α/β. (b) In the palindromic case we have l = 2, B0=A?1=B2? and B1= A0=A?0.
Ultimately, the perturbation of the palindromic eigenvalues is controlled byθ1
in (2.2), which is in turn dominated by the error term involving k[δ A0, δ A1]k.
The condition of the eigenvalues will be poor when κ is large or when deflating subspaces for different eigenvalues are getting “close” to each other, making the resolution of the spectrum more and more difficult. Note also that the perturbation in δ A0 may be nonsymmetric, pushing a pair of reciprocal
palindromic eigenvalues to ones that are not reciprocal (or approximately reciprocal whenδ A0is small). For a symmetricδ A0, we only have to consider
the perturbation of half of the eigenvalues, owing to the palindromic structure. (c) Based on Theorem 2.1, we can consider the perturbation of a cluster of
eigenvalues; for details, see [5, Section 5.2]. A cluster, to be defined later in (2.3), can be one simple eigenvalue, a group of multiple eigenvalues or a group of neighbouring eigenvalues. For(α, β) 6∈ σ (L), assume the following decomposition of the resolvent:
L(α, β)−1=X1T1(α, β)−1Z1+X2T2(α, β)−1Z2,
where ([X1, X2], T1⊕T2, [Z1, Z2]) is a resolvent triple [10] appropriately
partitioned into two parts. The eigenvalues in T1form a cluster when
X1T1(α, β) −1Z 1 X2T2(α, β) −1Z 2 ≤ X1T1(α, β) −1Z 1 (2.3) for some small constant. Consequently,
L(α, β) −1 ≤(1 + ) X1T1(α, β) −1Z 1 .
Arguments and techniques similar to those used in proving Theorem2.1can then be applied to L(α, β) + δL(α, β), so that (1 + ) X1T1(α, β) −1 Z1 kδL(α, β)k ≥ L(α, β) −1δL(α, β) ≥1
and (1 + )κ1kδL(α, β)k ≥ T1(α, β) −1 −1 , κ1≡ kX1k kZ1k.
Replacing kT1(α, β)−1kby an upper bound (as inAppendix A where T1 is in
Jordan or Kronecker form) yields results similar to those in Theorem2.1, but for the cluster in T1rather than the whole spectrumσ (L). Here p will be the size of
the largest Jordan block associated with the cluster in T1. Ignoring higher-order
terms in, the perturbation results will then involve κ1 instead ofκ. The price
to pay for the sharper result is the restriction that the perturbationδL has to be small (in the sense of (2.3)), whereas in Theorem2.1it can be arbitrary.
(d) In (c) above, when T1 contains a simple eigenvalue, κ1 will be the product
of the norms of the corresponding left- and right-eigenvectors. Similarly, for a group of multiple eigenvalues, the corresponding condition number will be the product of the norms of the corresponding left- and right-eigenvectors (or deflating subspaces). Analogous condition numbers can be obtained for clusters of eigenvalues.
(e) Obviously, for large perturbations withθ1> 1, we have max{θ1, θ11/p} =θ1. On
the other hand, whenθ1< 1, which is usually the case in (c) above, the maximum
occurs at θ11/p. Furthermore, when the perturbation is asymptotically small, p in (2.1) equals the size of the Jordan block associated with(αk, βk), where the
minimum in s(α,β)≡mini{|αβi−βαi|}occurs at i = k. (In fact, a perturbation
can be considered “small” when this correct pairing occurs; see the proof in Theorem 3.1.) Notice that the pth root is a common feature in perturbation results for multiple eigenvalues.
(d) A feature of the Bauer–Fike-type perturbation result is that one starts with a perturbed eigenvalue(α, β) whose spectral variation from a nearby unperturbed eigenvalue(αi, βi) is bounded. As the size of the perturbation is unrestricted,
there may well be unperturbed eigenvalues that are not paired up with any perturbed eigenvalues.
3. Bauer–Fike theorem for palindromic pencils
For the pencilλZ − Z?, we can work from the Kronecker canonical form Q?1λZ − Z? Q2=λ3+−3−, Q1= [P1, P2], Q2= [P2, P1], where 3+= 3 I , 3−= I 3 , 3 = diag{J1, . . . , JN},
with Ji being the Jordan block forλi on or inside the unit circle.
We have the following Bauer–Fike perturbation result.
THEOREM3.1. Consider the palindromic pencil L ≡β Z − αZ? with the above Kronecker canonical form. Let ˜Z = Z +δZ, ˜L ≡ β ˜Z − α ˜Z?, (αi, βi) ∈ σ (L) and
(α, β) ∈ σ( ˜L), with the scaling |αi|2+ |βi|2=1 = |α|2+ |β|2. The spectral variation
of eL from L is defined as sL(eL) ≡ max
(α,β){s(α,β)}, s(α,β)≡mini
{|αβi−βαi|}. Then, for any H¨older norm k · k,
s(α,β)≤maxnθ2, θ21/po, θ2≡c2κ2kδZk,
whereκ2 is defined as in(3.2), p is the size of the largest Jordan block in 3 and
c2=2 p 2(|α|2+ |β|2) = 2√2. Also, sL(eL) ≤ maxnθ2, θ 1/p 2 o.
PROOF. Applying the techniques in [5], we consider the singular matrix
Q?1hβ(Z + δZ) − α(Z + δZ)? i Q2 =(β3+−α3−) h I +(β3+−α3−)−1Q?1(βδZ − αδZ?)Q2i,
which implies that κ2 (β 3+−α3−) −1 β δZ −αδZ ? ≥1 (3.1) where κ2≡ kQ1k kQ2k. (3.2)
FromAppendix A, we have the upper bound (β 3+−α3−) −1 ≤c0max n |zi|−1, |zi|−pio,
where zi ≡αβi −βαi, pi is the size of the Jordan block associated with(αi, βi) and
c0≤2. Substituting this bound into (3.1), we obtain
κ2c2kδZk max n |zi|−1, |zi|−pi o ≥1 ⇒ minn|zi|, |zi|pi o ≤κ2c2kδZk.
The conclusions of the theorem then follow. 2
COMMENTS3.2.
(a) Note that for the 2-norm or the (Frobenius) F-norm, θ2≤
q
2(|α|2+ |β|2)pkδZk.
With the scaling |α|2+ |β|2=1 = |αi|2+ |βi|2, s(α,β) becomes the chordal
metric [5,11] andθ2≤
√
2 p kδZk. Although the F-norm is not a H¨older norm, the corresponding results can be deduced from the 2-norm results.
(b) Comments similar to those labelled (c)–(f) after Theorem 2.1 apply for Theorem3.1. The proof of the results for clusters of eigenvalues, using the partitioning of an appropriate resolvent, is similar and will not be repeated here.
(c) Details associated with palindromic “linearizations” can be found in [16]. Obviously, results in this section are applicable to general palindromic pencils which may not be linearizations of matrix polynomials.
4. Bauer–Fike theorem for anti-triangular form
From [18], we have the following anti-triangular canonical form for? = T . THEOREM4.1. Let Z −λZ? be a regular n × n palindromic pencil. There exists a unitary U ∈Cn×n
such that U?Z U =(mi j) with mi j=0 for i + j ≤ n + 1 (that is,
U?Z U is anti-triangular, with zero elements in the upper left corner).
Note that the result for ? = H can easily be obtained by extending the proofs of [18, Lemma 2.2 and Theorem 2.3]. Note also that we are only interested in the case where Z −λZ?is regular, which does not hold in [18].
The eigenvalues of the palindromic pencil Z −λZ?are m1,n m?n,1, m2,n−1 m?n−1,2, . . . , mi,n−i+1 m?n−i +1,i, . . . , mn−i +1,i m?i,n−i+1, . . . , mn−1,2 m?2,n−1, mn,1 m?1,n.
Note that n will be even when considering a linearization of a palindromic quadratic pencil [16], but the results in this section hold for any n.
Let N be the strict lower right triangular part of U?Z U. Reorganize the anti-triangular form in Theorem4.1into upper triangular form
PnU?(Z − λZ?)U = (D1+N1) − λ(D2+N2) (4.1)
with the order-reversing permutation matrixPn= [en, en−1, . . . , e1]; here
D2=diag{m1,n, m2,n−1, . . . , mn−1,2, mn,1},
D1=PnD2Pn=diag{mn,1, mn−1,2, . . . , m2,n−1, m1,n},
where N1=PnN and N2=PnN?are strictly upper triangular.
Using the Schur-like form in (4.1), we can prove the following perturbation result for a palindromic pencil.
THEOREM4.2. Consider the palindromic pencil L ≡β Z − αZ?, with N being the strict lower right triangular part of its anti-triangular canonical form. Let ˜Z =Z +δZ, ˜L ≡ β ˜Z − α ˜Z?, (αi, βi) ∈ σ (L) and (α, β) ∈ σ ( ˜L). Assume the scaling
|α|2+ |β|2=1 = |αi|2+ |βi|2. Then, for any H¨older norm k · k, s(α,β)≤
√
2 c0 maxnθ3, c3θ31/po, θ3≡ kδZk
for some p ≤ n and c0≡min{2, p}, c3≡
√ 2 kN k1−1/p. Also, sL(eL) ≤ √ 2c0maxnθ3, c3θ31/po.
PROOF. Consider the singular matrix β(Z + δZ) − α(Z + δZ)? =(U?)−1Pnhβ(D1+N1+PnU?δZU) − α(D2+N2+PnU?δZ?U) i UH =(U?)−1Pn[β(D1+N1) − α(D2+N2)] ×nI + [β(D1+N1) − α(D2+N2)]−1PnU?(βδZ − αδZ?)U o UH. It is easy to see that
[β(D1+N1) − α(D2+N2)] −1 β δZ −αδZ ? ≥ [β(D1+N1) − α(D2+N2)] −1P nU?(βδZ − αδZ?)U ≥1. (4.2) Note that [β(D1+N1) − α(D2+N2)] is assumed to be nonsingular, otherwise the
results in the theorem become trivial.
With z ≡ min(αi,βi)kβ D1−αD2k =min(αi,βi)|βαi −αβi|, ˜D ≡β D1−αD2and
˜ N ≡β N1−αN2, we have M ≡ [β(D1+N1) − α(D2+N2)]−1 =hI −(β D1−αD2)−1(β N1−αN2) i−1 (β D1−αD2)−1= I − ˜D−1N˜ −1 ˜ D−1. As ˜D−1N˜ is nilpotent, there exists some p ≤ n such that
I − ˜D−1N˜ −1 =I + ˜D−1N + · · · +˜ ˜ D−1N˜ p−1 and we obtain kMk ≤ ˜ N −1 η−1≡z−11 + ˜ N z −1+ · · · + ˜ N p−1 z−p+1. (4.3) With x ≡ k ˜N k−1z, we have the polynomial P(x) ≡ xp−η(1 + x + · · · + xp−1) = 0, as inAppendix Aor [5]. The only positive root x of P satisfies
x ≡ ˜ N −1 z ≤ c0 maxnη, η1/po. As (4.2) and (4.3) imply √ 2 kδZk ≥ kβδZ − αδZ?k ≥ kMk−1≥ ˜ N η,
this and the upper bound in (4.3) then lead to z ≤ c0 ˜ N maxnη, η 1/po ≤ √ 2 c0 max n kδZk, ˜ N 1−1/p kδZk1/po. As k ˜N k ≤ √
2 kN k, the results in the theorem then follow. 2 With the chosen scaling |α|2+ |β|2=1 = |αi|2+ |βi|2, s(α,β) equals the chordal
5. Bauer–Fike theorem for semi-Schur anti-triangular form
A refinement of Theorem4.2, with a smaller value for p, can be proved. We first refine the decomposition in Theorem4.1.
THEOREM5.1. Let Z −λZ? be a regular palindromic pencil. There exist nonsingular U , V ∈Cn×nsuch that
V?Z U =anti-diag {M1, . . . , Mr}, Mj=Dj+Nj, (5.1)
where Mj is anti-triangular with anti-diagonal elements in Dj =anti-diag
{λj, . . . , λj}.
PROOF. The proof is similar to the standard transformation of a Schur decomposition to the corresponding Jordan canonical form. It suffices to show that it is possible to transform the anti-triangular form ( ˜U?Z ˜U, ˜U?Z?U˜) in Theorem 4.1to anti-block-diagonal form, so that
I 0 P? I ˜ U?Z ˜U I Q 0 I = I 0 P? I 0 T1 T2 T12 I Q 0 I = T1 T2 , I 0 P? I ˜ U?Z?U˜ I Q 0 I = I 0 P? I 0 T2? T1? T12? I Q 0 I = T2? T1?
when T1 and T2 have nonintersecting spectra. Multiplying out the above equations
produces
φ(P, Q) ≡ (T2Q + P?T1, T1?Q + P?T2?) = −(T12, T12?), (5.2)
which is uniquely solvable [3]. 2
Similar to Theorem5.1, but with p being bounded by the maximum size of Mj, we
now have the following refined version of Theorem4.2.
THEOREM5.2. Consider the palindromic pencil L ≡β Z − αZ?with its semi-Schur anti-triangular canonical form (5.1). Let ˜Z = Z +δZ, ˜L ≡ β ˜Z − α ˜Z?, (αi, βi)
∈σ(L) and (α, β) ∈ σ ( ˜L). Assume the scaling |α|2+ |β|2=1 = |αi|2+ |βi|2. Then,
for any H¨older norm k · k, s(α,β)≤
√
2 c0 maxnθ4, c4θ41/po, θ4≡κ4kδZk
withκ4 ≡ kU k kV k, c0≡min{2, p} and c4≡
√
For sufficiently small perturbations, p is the size of the Schur block Mj associated
with(αi, βi). In general we have p = p∗, the maximum size of the Schur blocks Mj.
Also, sL(eL) ≤ √ 2 c0 maxnθ4, c4θ1/p ∗ 4 o.
PROOF. The proof is exactly the same as that for Theorem4.2, except that the kMk in (4.3) now equals the maximum of the norms of its diagonal blocks. The same argument can be followed in a similar fashion as in the proof of Theorem4.2, using the diagonal block at which the maximum occurs.
When the perturbation is small enough, this maximum (nearly infinite) occurs at the same block associated with(αk, βk) at which mini{|αβi−βαi|}occurs. This gives a
sharper perturbation result, with a smaller p which is just the size of the diagonal block
associated with(αk, βk). 2
Comments similar to those labelled (c)–(f) after Theorem2.1apply for the above theorem. Like Theorem 3.1, when we consider sufficiently small perturbations there will be a one-to-one correspondence between the original and the perturbed eigenvalues. The above perturbation bounds can be proved for a particular eigenvalue (αi, βi) with the condition number κ4 replaced by kUjk kVjk. In addition, instead
of considering one particular eigenvalue, we can consider a group of neighbouring eigenvalues together. This will increase p or the size of the corresponding semi-Schur block Mj, but will improve the condition of the linear operatorφ in (5.2) as well asκ4.
6. Perturbation by differentiation The results in this section are quoted from [7].
Without establishing differentiability or the existence of asymptotic expansions (which can be achieved by using the implicit function approach), perturbation results can be obtained via simple differentiation. See [1] for more details of this approach.
For some fixed z 6= 0, consider the palindromic eigenvalue problem P(λ, ρ)x(ρ) = 0, P(λ, ρ) ≡ λ(ρ)2A?1(ρ) + λ(ρ)A0(ρ) + A1(ρ)
with the scaling z?x(ρ) − 1 = 0, where ρ is the perturbation parameter, A0(0) = A0
and A1(0) = A1. We shall use notation(·)ρand(·)λto denote the corresponding partial
derivatives. For a simple eigenvalueλ, differentiation produces, at ρ = 0, λρ = −y ?Pρx y?Pλx = − y?Pρx y?(2λA?1+ A0)x (6.1) and P xρ= −(λρPλ+Pρ)x, z?xρ =0.
Upon choosing z = y(0) (the left-eigenvector corresponding to λ(0)) we obtain, at ρ = 0,
xρ = −P†(λρPλ+Pρ)x, where P†denotes the Penrose generalized inverse [11] of P.
The usual conclusions can be drawn: the right-eigenvector x will be rotated through a big angle, even for a small perturbation, when kP†k is large, that is, when the separation betweenλ and the other eigenvalues is fine. This happens, of course, when the assumption of simplicity for the eigenvalue is close to collapsing.
Note that for palindromic eigenvalue problems with? = T , the eigenvalues λ = ±1 may be multiple and nondifferentiable; thus a more sophisticated approach, like the one in [4], is required.
For perturbation results obtained through the application of Sun and Stewart’s approach [9, 19] in terms of the implicit function theorem, see [7]. Asymptotic perturbation series for the eigenvalues and the deflating subspaces have been derived.
7. Conclusions
Bauer–Fike-type perturbation results for general matrix polynomials, palindromic linearizations and (semi-Schur) anti-triangular canonical forms have been discussed for perturbations of arbitrary size. These perturbation results complement the ones for asymptotic perturbations given in [7]. Consistent results for simple eigenvalues and their corresponding eigenvectors were obtained using simple differentiation. These results indicate, not surprisingly, that the perturbations of an eigenvalue λ and its corresponding deflating subspaceSλare proportional, respectively, to the size of the perturbation and the reciprocal of the gap betweenSλand other deflating subspaces. Condition numbers are typically proportional to the products of the norms of the left-and right-eigenvectors or deflating subspaces.
Appendix A. Bounding k(β3+−α3−)−1k For any H¨older norm, given
3+= 3 I , 3−= I 3 , 3 = diag {J1, . . . , JN}
with the eigenvalues of Ji all on or inside the unit circle, we have
k(β3+−α3−)−1k =max i
kMik,
where Mi ≡(αI − β Ji)−1∈Rpi×pi.
Thus it is sufficient to consider the bound for kMik. Let zi ≡αβi −βαi, withαiand
βi being diagonal elements of I and Ji, respectively. (Here,αi =1 and |βi| ≤1; the
case where Mi ≡(α Ji −β I )−1can be treated similarly, and the symmetric notations
We have the Toeplitz matrix Mi ≡ zi −β zi −β 0 ... ... 0 zi −β z −1 = zi−1 βzi−2 β2zi−3 · · · βpi−1z−pi i zi−1 βz−i 2 · · · βpi−2z−pi+1 i z−i 1 ... ... 0 ... βzi−2 zi−1 . We then have kMik ≤c1·max n |zi|−1, |zi|−pio.
Finally, pi and |zi| can be replaced by p ≡ max pi and z ≡ mini |αβi−βαi|,
respectively, with c1=p.
Alternatively, potentially sharper bounds can be obtained, with c1 replaced by c0.
If Mi−1=zi Iqi −z
−1
i N(i) for a nilpotent N(i)such that N(i)
qi =0, then Mi =z−i 1 qi−1 X j =0 z−i 1 h N(i) ij , kMik ≤η−1≡ |zi|−1 qi−1 X j =0 |zi|−j.
For simplicity, let x = |zi| and m = qi. The above definition of η leads to the
polynomial
Pm(x) ≡ xm−η(1 + x + · · · + xm−1).
Descartes’ sign rule (La G´eom´etrie 1637) then implies that Pm(x) has at most one
positive real root. As Pm(0) = −η < 0 and Pm(x) > 0 as x → ∞, any positive
number x∗ for which Pm(x∗) > 0 is an upper bound of the unique real positive root
of Pm(x). Simple inspection leads to the upper bounds x∗=c0η when η > 1, and
x∗=c0η1/m whenη ≤ 1, with c0=min{2, m}. Consequently, c0=1 when m = 1
(with x =η), and c0=2 when m> 1.
The details are as follows. When c0η > η ≥ 1 and m > 1,
Pm(c0η) = (c0η)m−η(c0η) m− 1 c0η − 1 = c m+1 0 ηm+1−c m 0ηm−c m 0ηm+1+η c0η − 1 = (c m 0ηm+1−c m 0ηm) + (c m+1 0 ηm+1−c m 0ηm+1−c m 0ηm+1) + η c0η − 1 ≥0,
because cm+10 ηm+1−cm0ηm+1−cm0ηm+1=(c0−2)cm0ηm+1=0. Thus c0η is an
upper bound of the root x of Pmwhenη ≥ 1.
Whenη < 1 and m > 1, Pm(c0η1/m) ≥ cm0η − η(1 + c0+ · · · +cm0) = c m 0η − η c0m−1 c0−1 =η > 0.
Thus c0η1/mis an upper bound of the root x of Pmwhenη < 1.
Finally,
|zi| ≤c0max{η, η−qi} ⇒ kMik ≤η−1≤c0 max
n
|zi|−1, |zi|−qio,
and the result follows, with c1replaced by the sharper c0.
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