Eric King-wah Chu · Wen-Wei Lin · Chern-Shuh Wang
Perturbation of Palindromic
Eigenvalue Problems
Received: date / Revised version: date
Abstract We investigate the perturbation of the palindromic eigenvalue problem for the matrix quadratic P (λ) ≡ λ2AT
1 + λA0+ A1, with A0, A1∈
Cn×nand AT
0 = A0. The perturbation of palindromic eigenvalues and
eigen-vectors, in terms of general matrix polynomials, palindromic linearizations, (semi-Schur) anti-triangular canonical forms, differentiation and Sun’s im-plicit function approach, are discussed.
Keywords Anti-triangular form, eigenvalue, eigenvector, implicit function theorem, palindromic linearization, matrix polynomial, palindromic eigenvalue problem, perturbation.
Author, Journal Volume, (year) page numbers.
Eric King-wah Chu
School of Mathematical Sciences, Building 28, Monash University, VIC 3800, Aus-tralia.
Tel.: +61-3-99054480 Fax: +61-3-99054403
E-mail: [email protected] Wen-Wei Lin
Department of Mathematics, National Tsinghua University, Hsinchu 300, Taiwan. Tel.: +886-3-5731057
Fax: +886-3-5723888
E-mail: [email protected] Chern-Shuh Wang
Department of Mathematics, National Cheng Kung University, Tainan 701, Tai-wan.
Tel.: +886-6-2757575 Fax: +886-6-2743191
1 Introduction
Consider the matrix quadratic
P (λ) ≡ λ2AT1 + λA0+ A1 (1)
where A0, A1 ∈ Cn×n and AT0 = A0, and the corresponding palindromic
quadratic eigenvalue problem
P (λ)x = 0 , x 6= 0 (2)
In general, an eigenvalue problem is described as palindromic if the spectrum contains both λ and λ−1. In other words, the eigenvalue problem of the origi-nal matrix polynomial P (λ) (in one of its equivalent forms) has a palindromic linearization [3, 8] of the form λZ ± ZT. (We can transform λZ − ZT back
to the form ν(−Z) + (−Z)T with ν = −λ. Similarly, λ2AT
1 + λA0+ A1 and
ν2AT
1 − νA0+ A1 define equivalent palindromic eigenvalue problems.) It is
clear that (2) is palindromic, after checking its transpose.
A great foundation for the solution of palindromic eigenvalue problems has been laid by Hilliage, Mackey, Mehl and Mehrmann in [6, 8, 9]. There has been much recent interest in quadratic eigenvalue problems [14]. An im-portant example of palindromic eigenvalue problems can be found in the vibration analysis of fast trains; see [6, 7] for general introductions and [5] for details. For general perturbation of eigenvalues for polynomial eigenvalue problems, see [1, 13]. On results for general matrix polynomials, see the mas-terpiece [3].
This paper is organized as follows. In Sections 2–5, the perturbation of palindromic eigenvalues, in terms of general matrix polynomials, palindromic linearizations, the anti-triangular canonical form [8–10] and the semi-Schur anti-triangular canonical form, is discussed using the Bauer-Fike technique [1]. The perturbation result for a simple eigenvalue and its corresponding eigenvector is obtained by differentiation in Section 6. Sun’s implicit function approach [12] is then applied to palindromic linearizations, general matrix quadratics and palindromic eigenvalue problems in Section 7, to obtain per-turbation results for eigenvalues and the corresponding deflating subspaces. The paper is concluded in Section 8.
Note that the Bauer-Fike technique allows perturbations of arbitrary size and the clustering of eigenvalues (and the corresponding deflating subspaces) may vary greatly. Consequently, it is meaningless to talk about the perturba-tion of eigenvectors or deflating subspaces in Bauer-Fike type perturbaperturba-tion theorems in Sections 2–5.
2 Bauer-Fike Theorem for General Matrix Polynomials
From [1, Theorem 4.2], we have the following theorem on the perturbation of eigenvalues of a general matrix polynomial:
Theorem 21 Consider a regular matrix polynomial L(α, β) ≡ l X j=0 Bjαjβl−j ,
and its perturbation
e L(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 constructed using some finite and infinite Jordan pairs JF and J∞. For (αi, βi) ∈ σ(L) and (α, β) ∈ σ( eL), the
spectral variation of eL from L is defined as sL( eL) ≡ max
(α,β){s(α,β)} ,
s(α,β)≡ min
i {|αβi− βαi|} .
Let p be the maximum dimension of the Jordan blocks in JF or J∞.
Then for k · kτ (τ = 1, 2, ∞), we have
s(α,β)≤ max{θ1, θ 1/p 1 } , θ1≡ pF κ∆ , (3) where c1= 1 (for τ = 1, 2) or √ l (for τ = ∞), and F ≡ c1 r l + 1 2 , κ ≡ kXk · kZk , ∆ ≡ k[δB0, · · · , δBl]k . Also, we have sL( eL) ≤ max{θ1, θ 1/p 1 } . (4)
Note that we use the representation (α, β) for λ = α/β in the above theorem. In our palindromic case, we have l = 2 and B0 = AT1 = B2T and
B1= A0= AT0. Ultimately, the perturbation of the palindromic eigenvalues is
controlled by θ1in (4), which is in turn dominated by the error term involving
k[δA0, δA1]k. Note also that the perturbation in δA0 may be nonsymmetric,
pushing a pair of reciprocal palindromic eigenvalues to one which is only approximately reciprocal.
For a symmetric δA0, we only have to consider the perturbation of half
of the eigenvalues, because of the palindromic structure.
The unstructured perturbation result in Theorem 21, for general matrix polynomials, may not be applicable or satisfactory for palindromic eigenvalue problems. However, it serves as a reference for the structured perturbation results are in later Sections.
3 Bauer-Fike Theorems for Palindromic Linearizations
For the linearization λZ + ZT, we can work from the Kronecker canonical form QT1 λZ − ZT Q2= λΛ+− Λ− , Q1= [P1, P2] , Q2= [P2, P1] where Λ+= ΛI , Λ−= IΛ , Λ = diag{J1, · · · , JN}
with Ji being the Jordan block for λi inside the unit circle.
Applying the techniques in [1], we consider the singular matrix QT1 β(Z + δZ) − α(Z + δZ)T Q 2 = (βΛ+− αΛ−)I + (βΛ+− αΛ−)−1QT1(βδZ − αδZ T)Q 2 which implies κ2k(βΛ+− αΛ−)−1k kβδZ − αδZTk ≥ 1
where κ2≡ kQ1k kQ2k. Estimation of k(βΛ+−αΛ−)−1k, with Λ±containing
various Jordan blocks, produces the following theorem.
Theorem 31 Consider the palindromic linearization M ≡ βZ − αZT with
the above Kronecker canonical form. Let ˜Z = Z + δZ, ˜M ≡ β ˜Z − α ˜ZT,
(αi, βi) ∈ σ(M ) and (α, β) ∈ σ( ˜M ). 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 Holder norm k · k, we have s(α,β)≤ max{θ2, θ
1/p
2 } , θ2≡ c2κ2kδZk (5)
for some p ≤ n and c2=p2(α2+ β2) p.
Also, we have
sL( eL) ≤ max{θ2, θ 1/p
2 } . (6)
We have the following three cases:
(i) for small perturbations, we have p being the size of the Jordan block associated with (αi, βi);
(ii) for large perturbations, we have p = 1; and
(iii) for general perturbations, we have p being the size of the largest Jordan block in Λ.
Note that for the 2- or F-norm, we have θ2 ≤ p2(α2+ β2) p kδZk.
With scaling,p|α|2+ |β|2 = 1, s
(α,β) becomes the chordal metric and θ2 ≤
√
2 p kδZk.
From [1], when we consider asymptotically small perturbations, there will be a 1-1 correspondence between the original and perturbed eigenvalues. The above perturbation bounds can be proved for a particular eigenvalue (αi, βi)
with the condition number κ2replacable by the product of the norms of the
4 Perturbation of Palindromic Linearizations in Anti-Triangular Form
From [10], we have the following anti-triangular canonical form:
Theorem 41 Let Z + λZT be a regular n × n palindromic linearization.
There exists a unitary U ∈ Cn×n such that UTZU = (m
ij) with mij = 0
(i + j ≤ n + 1) (i.e., UTZU is anti-triangular, with zero elements on the
upper left corner).
The palindromic eigenvalues are: −m1n mn1 , −m2,n−1 mn−1,2 , · · · , −mi,n−i+1 mn−i+1,i , · · · , −mn−i+1,i mi,n−i+1 , · · · , −mn−1,2 m2,n−1 , −mn1 m1n
Let N be the strict lower right triangular part of UTZU . Reorganize the
anti-triangular in Theorem 41 in lower triangular form:
UT(Z + λZT)U Pn= (D1+ N1) + λ(D2+ N2) (7)
with the order-reversing permutation matrix Pn= [en, en−1, · · · , e1], we have
D1= diag{m1n, m2,n−1, · · · , mn−1,2, mn1}
D2= PnD1Pn = diag{mn1, mn−1,2, · · · , m2,n−1, m1n}
with N1= N Pn and N2= NTPn being strictly lower triangular.
Using the Schur form in (7), we can prove the following perturbation result for a palindromic linearization.
Theorem 42 Consider the palindromic linearization M ≡ βZ − αZT with
N being the strictly lower right triangular part of its anti-triangular canonical form. Let ˜Z = Z + δZ, ˜M ≡ β ˜Z − α ˜ZT, (α
i, βi) ∈ σ(M ) and (α, β) ∈ σ( ˜M ).
Assume the scaling α2+ β2= 1 = α2
i + βi2. Then for any Holder norm k · k,
we have
s(α,β)≤ c0max{θ3, θ 1/p
3 } , θ3≡ c3kδZk (8)
for some p ≤ n and c0≡ min{2, p}, c3=
√ 2 pkN k. Also, we have sL( eL) ≤ c0max{θ3, θ 1/p 3 } . (9)
Proof. Consider the singular matrix
β(Z + δZ) − α(Z + δZ)T = Uβ(D1+ N1+ UTδZU Pn) − α(D2+ N2+ UTδZTU Pn) PnUH = U [β(D1+ N1) − α(D2+ N2)] · n I + [β(D1+ N1) − α(D2+ N2)] −1 UH(βδZ − αδZT)U Pn o PnUH
It is easy to see that [β(D1+ N1) − α(D2+ N2)] −1 βδZ − αδZT ≥
[β(D1+ N1) − α(D2+ N2)] −1 UH(βδZ − αδZT)U Pn ≥ 1 (10) Note that [β(D1+ N1) − α(D2+ N2)] is assumed to be nonsingular,
other-wise the results in the theorem become trivial.
With z ≡ min(αi,βi)kβD1− αD2k = min(αi,βi)|βαi− αβi| and
M ≡ [β(D1+ N1) − α(D2+ N2)]−1
=I − (βD1− αD2)−1(βN1− αN2)
−1
(βD1− αD2)−1
Using the Neumann series with ˜D ≡ βD1− αD2and ˜N ≡ βN1− αN2:
I − (βD1− αD2)−1(βN1− αN2) −1
= (I − ˜N )−1= I + ˜N + · · · + ˜Np−1 we obtain
kM k ≤ k ˜N k−1η−1 ≡ z−1(1 + k ˜N kz−1+ · · · + k ˜N kp−1z−p+1) With x ≡ k ˜N k−1z, we have the polynomial xp− η(1 + x + · · · + xp−1) = 0
(see the result in the Appendix), the technique in [1] bounds kM k via x ≡ k ˜N k−1z ≤ c0max{η, η1/p}
Together with the properties of norms, this completes the proof.
With scaling, s(α,β) equals the chordal metric. Note also that p is the
integer for which ˜Nk6= 0 (0 ≤ k < p) and ˜Np= 0.
5 For Palindromic Linearizations in Semi-Schur Anti-Triangular Form
A refinement of Theorem 42, with smaller values for p, can be proved. We first refine the decomposition in Theorem 41.
Theorem 51 Let Z + λZT be a regular palindromic linearization. There exists a nonsingular U , V ∈ Cn×n,
VTZU = anti-diag{m1, · · · , mr} , mj= Dj+ Nj (11)
where mj is anti-triangular with anti-diagonal block Dj= anti-diag{λj}.
Proof. The proof is similar to the standard transformation of a Schur de-composition to the corresponding Jordan canonical form. It is sufficient to show that it is possible to transform the anti-triangular form (UTZU, UTZTU )
in Theorem 41 to anti-block diagional form, so that I 0 PT I UTZU I Q 0 I = I 0 PT I 0 T1 T2 T12 I Q 0 I = T1 T2 I 0 PT I UTZTU I Q 0 I = I 0 PT I 0 T2T TT 2 T12T I Q 0 I = T2T TT 1
assuming that the spectra of T1and T2are nonintersecting. Multiplying out
the above equation produces the simultaneous linear matrix equation φ(P, Q) ≡ (T2Q + PTT1, T1TQ + PTT2T) = −(T21, T21T) (12)
which is uniquely solvable.
Similar to Theorem 51 but with p being the bounded by the maximum size of mj, we can now prove the following refined version of Theorem 42:
Theorem 52 Consider the palindromic linearization M ≡ βZ − αZT with
its semi-Schur anti-triangular canonical form (11). Let ˜Z = Z + δZ, ˜M ≡ β ˜Z −α ˜ZT, (α
i, βi) ∈ σ(M ) and (α, β) ∈ σ( ˜M ). Assume the scaling α2+β2=
1 = α2
i + βi2. Then for any Holder norm k · k, we have
s(α,β)≤ c0max{θ4, θ 1/p
4 } , θ4≡ c4kδZk (13)
for κ4≡ kU k kU−1k, and c0≡ min{2, p}, c4=
√ 2 pκ4maxjkNjk. Also, we have sL( eL) ≤ c0max{θ4, θ 1/p 4 } . (14)
We have the following three cases:
(i) for small perturbations, we have p being the size of the Schur block mj
associated with (αi, βi);
(ii) for large perturbations, we have p = 1; and
(iii) for general perturbations, we have p being the size of the largest Schur block mj.
Similar to Theorem 31, when we consider asymptotically small perturba-tions, there will be a 1-1 correspondence between the original and perturbed eigenvalues. The above perturbation bounds can be proved for a particular eigenvalue (αi, βi) with the condition number κ4 replacable by kUjk kVjk.
In addition, we can consider a group of neighbouring eigenvalues together, instead of one particular eigenvalue. This will increase p, or the size of the corresponding semi-Shcur block mj, but will improve the conditioning of the
linear operator φ in (12) as well as κ4.
6 Perturbation by Differentiation
Without establishing the existence of asymptotic expansions (which can be achieved using the implicit function approach in Section 7), perturbation results can be obtained by simple differentiation.
Consider the palindromic eigenvalue problem
with ρ being the perturbation parameter and A0(0) = A0, A1(0) = A1. For
a simple eigenvalue λ, differentiation produces λρ= − yTP ρx yTP λx = − y TP ρx yT(2λAT 1 + A0)x (15) and P xρ= −(λρPλ+ Pρ)x , zTxρ= 0
Choosing z = y(0), we have
xρ= −P†(λρPλ+ Pρ)x
The usual conclusion can be drawn — the right-eigenvector x will be rotated through a big angle, even for a small perturbation, when kP†k is
large or when the separation between λ and other eigenvalues is fine. This happens, of course, when the assumption of simplicity for the eigenvalues is nearly violated.
7 Sun’s Implicit Function Approach
In this Section, we shall abbreviate the development of Sun’s approach [2, 12], by establishing the functions to which the implicit function theorem can be applied. Detailed and subtle argument of this approach can be found in [2, 12].
7.1 Palindromic Linearizations
Using the anti-triangular form in (41), Sun’s approach [2] can be applied to obtain the power series of eigenvalues and deflating subspaces. Without loss of generality, we shall use an upper-triangular form in this section (with the help of the reordering operator Pr), so that Sun’s approach can be followed
faithfully.
Assume that U in (41) is further organized to reflect the symmetry of the eigenvalue pairs {λj, λ−1j }, so that
U = [U1, U0, U−1] = [U1, U2] = [U−2, U−1]
and
U Pn = [U−1Pp, U0Pn−2p, U1Pp] = [U−1Pp, U−2Pn−p] = [U2Pn−p, U1Pp]
The anti-triangular form now becomes
UT(Z − λZT)U Pn = Λα1 ∗ 0 Λα2 − λ Λβ1 ∗ 0 Λβ2 = Λα1 ∗ Λα0 0 Λα,−1 − λ Λβ1 ∗ Λβ0 0 Λβ,−1
with Λα1 and Λβ1 being p × p (2p < n),
Λβ,−1= PpΛα1Pp , Λβ,1= PpΛα,−1Pp , Λβ0= P2n−pΛα0P2n−p
and {U−1, U1} spanning the deflating subspaces of Z − λZT corresponding
to (Λα1, Λβ1).
Assume that (Λα1, Λβ1) and (Λα2, Λβ2) have nonitersecting spectra.
Fol-lowing the usual step in Sun’s approach [2], we assume that Z(ρ) is dependent on a perturbation parameter ρ. We then have
M ≡ UTZ(ρ)U Pn = M11 M12 M21 M22 ≡ U T 1ZU−1PpU1TZU−2Pn−p UT 2ZU−1PpU2TZU−2Pn−p and L ≡ UTZ(ρ)TU P n = L11 L12 L21L22 ≡ U T 1ZTU−1PpU1TZTU−2Pn−p UT 2ZTU−1PpU2TZTU−2Pn−p
From the (2,1)-block of Ip 0 −Ψ In−p (M − λL) Ip 0 Φ In−p we construct F (Φ, Ψ, ρ) G(Φ, Ψ, ρ) = −Ψ UT 1ZU−1Pp− Ψ U1TZU−2Pn−pΦ + U2TZU−1Pp+ U2TZU−2Pn−pΦ −Ψ UT 1ZTU−1Pp− Ψ U1TZTU−2Pn−pΦ + U2TZTU−1Pp+ U2TZTU−2Pn−pΦ
At ρ = 0, we have Φ = 0 = Ψ and F (0, 0, 0) = 0 = G(0, 0, 0). The implicit function theorem can then be applied to (F, G). Differentiation of F and G respect to Ψ and Φ at ρ = 0 yields (after stacking columns and apply Kronecker products) F G (Ψ,Φ) = −Λ T α1⊗ I I ⊗ Λα2 −ΛT β1⊗ I I ⊗ Λβ2
It is easy to see that the above operator is invertible when the spectra of (Λα1, Λβ1) and (Λα2, Λβ2) do not intersect (and it will be ill-conditioned
when the subspectra are close). Consequently, we have proved that the power series exists for Φ(ρ) and Ψ (ρ) within some small neighbourhood of ρ = 0 and
Φ(ρ) = Φρρ + · · · , Ψ (ρ) = Ψρρ + · · ·
Furthermore, the deflation subspace [2] is formed by span U I Ψ (p) , span U Pn I Φ(p) =span U1+ U2Ψ (p) , span (U−1Pp+ U−2Pn−pΦ(p))
Differentiation of M11≡ U1TZU−1Pp , L11≡ U1TZTU−1Pp also yields ∂M11 ∂ρ = U T 1ZρU−1Pp , ∂L11 ∂ρ = U T 1Z T ρU−1Pp
When p = 1 and (Λα1, Λβ1) = (λα1, λβ1) represents the finite eigenvalue
λ1= λα1/λβ1, the above derivatives translate to
∂λ1 ∂ρ = ∂λα1 ∂ρ λβ1− ∂λβ1 ∂ρ λα1 λ2 β1 = ∂λα1 ∂ρ λβ1 −λ1 ∂λβ1 ∂ρ λβ1 = y T 1Zρx1− λ1yT1ZρTx1 yT 1ZTx1
producing a result analogous to (15).
Lastly, differentiating F and G with respect to ρ at ρ = 0 produces Fρ = Λα2Φρ− ΨρΛα1+ U2TZρ(0)U−1Pp= 0
Gρ = Λβ2Φρ− ΨρΛβ1+ U2TZρ(0)TU−1Pp = 0
The derivatives Φρ(0) and Ψρ(0) can then be retrieved from the above
equa-tions, when (Λα1, Λβ1) and (Λα2, Λβ2) have nonintersecting spectra.
7.2 General Matrix Quadratics
We now apply Sun’s approach [2] to the general matrix quadratic Q2(λ) = λ2M + λD + K
similar to the development in the previous subsection. Assume the n × 2n matrices X = [xj] and Y = [yj] contain, respectively, the right- and
left-eigenvectors, with xjand yjcorresponding to λj = αj/βj. For the companion
linearization L(λ) ≡ 0 I −K −D − λ I M (16) it is easy to check that the right- and left-eigenvectors corresponding to λj= αj/βj are respectively βjxj αjxj ,βjyTjD + αjyjTM, βjyjT T
Notice that by using (αj, βj) to represent λj, we have avoided the difficulties
involving infinite eigenvalues. We can then scale the eigenvectors to satisfy the biorthogonality equations
ΛαYTM XΛβ+ ΛβYTM XΛα+ ΛβYTDXΛβ = Λβ (17)
Here Λαand Λβare block-diagonal matrices, with blocks being the usual
identity or Jordan matrices for the generalized eigenvalue sub-problems. Assume further that X, Y , Λαand Λβ are organized as follows:
X = [X1, X2] , Y = [Y1, Y2] , Λα= diag{Λα1, Λα2} , Λβ= diag{Λβ1, Λβ2}
where the spectra of (Λα1, Λβ1) and (Λα2, Λβ2) do not intersect.
Following the usual steps in Sun’s approach [2] in obtaining analytic power series for the eigenvalues and eigenvectors for generalized eigenvalue prob-lems, we assume the matrices M , D and K are dependent on a perturba-tion parameter ρ. We first transform the linearizaperturba-tion in (16) by pre-(post-)multiplying with its left-(right-)eigenvectors:
M ≡ M11M12 M21M22 ≡ΛβYTD + ΛαYTM, ΛβYT 0 I −K −D XΛβ XΛα and L ≡ L11 L12 L21 L22 ≡ΛβYTD + ΛαYTM, ΛβYT I M XΛβ XΛα
From the (2,1)-block of Ip 0 −Ψ In−p (M − λL) Ip 0 Φ In−p we construct F (Φ, Ψ, ρ) ≡ −Ψ (Λα1Y1TM X1Λα1− Λβ1Y1TKX1Λβ1) −Ψ (Λα1Y1TM X2Λα2− Λβ1Y1TKX2Λβ2)Φ +(Λα2Y2TM X1Λα1− Λβ2Y2TKX1Λβ1) +(Λα2Y2TM X2Λα2− Λβ2Y2TKX2Λβ2)Φ G(Φ, Ψ, ρ) ≡ −Ψ (Λβ1Y1TDX1Λβ1+ Λα1Y1TM X1Λβ1+ Λβ1Y1TM X1Λα1) −Ψ (Λβ1Y1TDX2Λβ2+ Λα1Y1TM X2Λβ2+ Λβ1Y1TM X2Λα2)Φ +(Λβ2Y2TDX1Λβ1+ Λα2Y2TM X1Λβ1+ Λβ2Y2TM X1Λα1) +(Λβ2Y2TDX2Λβ2+ Λα2Y2TM X2Λβ2+ Λβ2Y2TM X2Λα2)Φ
At ρ = 0, we have Φ = 0 = Ψ and F (0, 0, 0) = 0 = G(0, 0, 0). The implicit function theorem can then be applied to (F, G). Differentiation of F and G respect to Ψ and Φ at ρ = 0 yields (after stacking columns and apply Kronecker products) F G (Ψ,Φ) = −Λ T α1⊗ I I ⊗ Λα2 −ΛT β1⊗ I I ⊗ Λβ2
It is easy to see that the above operator is invertible when the spectra of (Λα1, Λβ1) and (Λα2, Λβ2) do not intersect (and it will be ill-conditioned
series exists for Φ(ρ) and Ψ (ρ) within some small neighbourhood of ρ = 0 and
Φ(ρ) = Φρρ + · · · , Ψ (ρ) = Ψρρ + · · ·
Furthermore, the deflation subspace [2] of the companion linearization is formed by span ˜ Y−T I Ψ (p) , span ˜ X I Φ(p) with ˜ Y =ΛβYTD + ΛαYTM, ΛβYT T , X =˜ XΛβ XΛα Differentiation of M11≡ Λα1Y1TM X1Λα1− Λβ1Y1TKX1Λβ1 L11≡ Λβ1Y1TDX1Λβ1+ Λα1Y1TM X1Λβ1+ Λβ1Y1TM X1Λα1 also yields ∂M11 ∂ρ = Λα1Y T 1 MρX1Λα1− Λβ1Y1TKρX1Λβ1 ∂L11 ∂ρ = Λβ1Y T 1 DρX1Λβ1+ Λα1Y1TMρX1Λβ1+ Λβ1Y1TMρX1Λα1
When p = 1 and (Λα1, Λβ1) = (λα1, λβ1) represents the finite eigenvalue
λ1= λα1/λβ1, the above derivatives translate to
∂λ1 ∂ρ = ∂λα1 ∂ρ λβ1− ∂λβ1 ∂ρ λα1 λ2 β1 = ∂λα1 ∂ρ λβ1 −λ1 ∂λβ1 ∂ρ λβ1 =−λ 2 β1yT1Kρx1+ λα12 yT1Mρx1− λ1(λ2β1yT1Dρx1+ 2λα1λβ1y1TMρx1) λ2 β1yT1Dx1+ 2λα1λβ1yT1M x1 = −y T 1(λ21Mρ+ λ1Dρ+ Kρ)x1 yT 1(2λ1M + D)x1 (19) producing the same formula in (15).
Lastly, differentiating F and G with respect to ρ at ρ = 0 produces Fρ = Λα2Φρ− ΨρΛα1+ Λα2Y2TMρ(0)X1Λα1− Λβ2Y2TKρ(0)X1Λβ1= 0
Gρ = Λβ2Φρ− ΨρΛβ1+ Λα2Λβ2Y2TDρ(0)X1Λβ1
+Λα2Y2TMρ(0)X1Λβ1+ Λβ2Y2TMρ(0)X1Λα1= 0
The derivatives Φρ(0) and Ψρ(0) can then be retrieved from the above
7.3 Palindromic Case
For palindromic eigenvalue problems, the perturbation results look almost identical to those for general matrix quadratics, as shown in the previous section. The only differences are M = AT1 = KH, D = A0 = DT, and the
palindromic properties of the eigenvalues and eigenvectors. For the spectrum, we have (Λα1, Λβ1) and (Λα2, Λβ2) = (Λβ1, Λα1) representing, respectively,
eigenvalues inside and outside the unit circle, and [Y1, Y2] = [X2, X1] (as
in Section 3.5.1). We shall not rewrite all the general results for palindromic quadratics, leaving the trivial substitution of symbols as an exercise.
For the simple palindromic eigenvalues λ1and λ−11 , (19) implies
∂λ1 ∂ρ = − yT 1(λ21KρT+ λ1Dρ+ Kρ)x1 yT 1(2λ1KT + D)x1 ∂λ−11 ∂ρ = y1T(KρT+ λ−11 Dρ+ λ−21 Kρ)x1 yT 1(2λ1KT + D)x1
Interestingly, the relative errors of the pair of reciprocal eigenvalues equal, asymptotically, to ρ λ±11 ∂λ±11 ∂ρ = ∓ρ y1T(λ1KρT + Dρ+ λ−11 Kρ)x1 yT 1(2λ1KT+ D)x1 (20) and are identical except of the opposite signs. Equation (20) can easily be understood from 1 λ + δλ = 1 λ 1 +δλ λ ≈ 1 λ 1 − δλ λ
After applying inequalities of norms, a condition number for both λ±11 can then be produced:
κ ≡ ky1k2kx1k2 |yT
1(λ1KT + D)x1|
p|λ1|2+ 1 + |λ1|−2 (21)
With appropriate scaling of the eigenvectors, κ can be interpreted as the product of the norms of the left- and the right-eigenvectors, or in terms of the angle between the eigenvectors, as in other algebraic eigenvalue problems.
8 Numerical Example
We shall consider the following small example to illustrate the results in the previous Section 7.3: A0= 1 2 2 1 , A1= 1 ρ 0 1
The parameter ρ can be used to vary the conditioning of the eigenvalue problem, but will be fixed to be 0.5 in the following calculations. The ma-trices are perturbed randomly to the magnitude of 0.5 × 10−7, with δ =
Table 1 Perturbation of a palindromic eigenvalue problem i 1 2, 3 4 λi −0.5488554937 0.6854143467 ± −1.8219731997 0.7281532623 i ˜ λi −0.5488554724 0.6854143626 ± −1.8219732705 0.7281532473 i
δλi 2.1312649867e-08 1.5930742392e-08 ± −7.0749180514e-08 1.4995688913e-08 i
ri −3.8831076873e-08 −4.1288419705e-16 ± 3.8831076399e-08 2.1878282362e-08 i
|ri| 3.8831076873e-08 2.1878282362e-08 3.8831076399e-08 r(e)i 3.8831078433e-08 4.7865922782e-16 ± 3.8831078433e-08
2.1878282332e-08 i
|r(e)i | 3.8831078433e-08 2.1878282332e-08 3.8831078433e-08 κi 1.2701114034 1.5072107111 1.2701114034 r(κ)i 8.8772549782e-08 1.0534425368e-07 8.8772549782e-08
k[δAT
1, δA0, δA1]k = 0.6989351449543007 × 10−7. The eigenvalues are λi
(i = 1, · · · , 4) with λ1= λ−14 and λ2= λ−13 . The numerical results are
sum-marized in the following table, with ˜λi denoting the perturbed eigenvalues,
δλi ≡ ˜λi− λi, ri ≡ δλi/λi the relative error in ˜λi, r (e)
i estimating ri using
(20), κi the individual condition numbers as in (21), and
r (e) i ≤ r (κ) i ≡ κiδ
estimating |ri|. All calculations were carried out using MATLAB 7.1 (R14)
on a Apple MacIntosh G4 Powerbook, with eps ≈ 2.2204 × 10−16.
It is obvious from Table 1 that (20) provides accurate approximations to the relative errors of palindromic eigenvalues with small perturbations and the condition number κ in (21) produces tight upper bounds for the (relative) errors. Also, the fact that the relative errors for reciprocal pairs of eigenvalues are negative of each other is confirmed by the example.
9 Conclusions
Bauer-Fike type perturbation results for general matrix polynomials, palin-dromic linearizations, the anti-triangular canonical form and the semi-Schur anti-triangular canonical form are discussed, for perturbations of arbitrary size. For asymptotic perturbations, perturbation results for eigenvalues and the corresponding deflating subspaces of palindromic linearizations and (palin-dromic) matrix quadratics are obtained, Sun’s implicit function approach. Consistent results for simple eigenvalues and the corresponding eigenvetors are obtained using simple differentiation. These results indicate, not surpris-ingly, that the perturbations of an eigenvalue λ and its corresponding deflat-ing subspace Sλ, respectively, are proportional to the size of the perturbation
condition number is typically the product of the norms of the left- and right-eigenvectors.
Appendix: Bounding the root of xm− η(1 + x + · · · + xm−1)
Consider 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 when η ≤ 1, with c0 = min{2, p}.
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, we have
Pm(c0η) = (c0η)m− η (c0η)m− 1 c0η − 1 = c m+1 0 η m+1− cm 0ηm− cm0ηm+1+ η c0η − 1 = (c m 0ηm+1− cm0 ηm) + (c m+1 0 ηm+1− cm0ηm+1− cm0ηm+1) + η c0η − 1 ≥ 0 as cm+10 ηm+1− cm 0ηm+1− cm0ηm+1 = (c0− 2)cm0ηm+1 = 0. Thus c0η is an
upper bound of the root x of Pmwhen η ≥ 1.
When η < 1 and m > 1, we have
Pm(c0η1/m) ≥ cm0 η − η(1 + c0+ · · · + cm0) = cm0 η − ηc m 0 − 1 c0− 1 = 0 as cm+10 − 2cm
0 = 0. Thus c0η1/mis an upper bound of the root x of Pmwhen
η < 1.
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