Condition numbers of multiple eigenvalue for
matrix polynomial eigenvalue problems
Yueh-Cheng Kuo
∗Abstract
In this paper, we consider the condition numbers of multiple eigenvalue for matrix polynomial eigenvalue problems.
1
Introduction
Consider the matrix polynomial has the form
P (λ) = λ`A`+ . . . + λA1+ A0, (1.1)
where A0, . . . , A` are n×n real or complex matrices. In this paper, we shall consider
only regular matrix polynomials P (λ), that is, det(P (λ)) ≡/ 0. The Polynomial Eigenvalue Problem (PEP): find a scalar λ ∈ C and nonzero vectors x, y ∈ Cnsuch
that
P (λ)x = 0, yHP (λ) = 0, (1.2) where x and y are the right and left eigenvectors corresponding to eigenvalue λ, respectively.. The most common way of solving the PEP is to convert it into a linear problem (λD − C)z = 0, (1.3) where D = diag(A`, In, . . . , In), C = A`−1 A`−2 · · · A0 −In 0 · · · 0 .. . . .. . .. ... 0 · · · −In 0 , z = λ`−1x λ`−2x .. . x . (1.4)
∗Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung, 811,
In this paper, we wish to analyze that how the multiple eigenvalues λ change of the PEP (1.2) when the coefficient matrices of P (λ) are subjected to small perturbations.
2
Preliminaries
In this section we introduce some notations and definitions, and give some prelim-inary results.
2.1
Definitions and notations
For any given a regular matrix polynomial P (λ) of degree `. Let λ be a multiple eigenvalue of PEP (1.2) with multiplicity mλ and let
xj,1, . . . , xj,nj, j = 1, . . . , q (2.5)
be the Jordan chains of matrix polynomial P (λ) corresponding to eigenvalue λ, that is, for each j ∈ {1, . . . , q} the vector [x>j,1, . . . , x>j,nj]> is the solution of the linear system P (λ) 0 · · · 0 P0(λ) P (λ) ... .. . . .. ... 1 (nj−1)!P (nj−1)(λ) 1 (nj−2)!P (nj−2)(λ) · · · P (λ) xj,1 xj,2 .. . xj,nj = 0, (2.6)
and the leading vector {x1,1, x2,1, . . . , xq,1} is linearly independent which is
eigen-vector corresponding eigenvalue λ. The n1, n2, . . . , nq are called the partial
multi-plicities or the length of Jordan chains for P (λ) corresponding to λ. Hereafter, we make an order relation on n1, n2, . . . , nq
n1 = · · · = nr1 > nr1+1 = · · · = nr2 > · · · > nrp−1+1 = · · · = nrp, rp = q. (2.7)
For convenience, we set r0 = 0, then the algebraic multiplicities of λ is
mλ = q X j=1 nj = p X i=1 (ri− ri−1)nri.
The eigenvalue λ is semi-simple (nondefective) if n1 = 1 and nonderogatory if
We denote Xλj = [xj,1, . . . , xj,nj] ∈ C n×nj, J λ,j = λ 1 . .. ... . .. 1 0 λ ∈ Cnj×nj (2.8)
for j = 1, 2, . . . , q. Then the equation (2.6) can be rewritten as
Jλ,j`> ⊗ A`+ J(`−1) > λ,j ⊗ A`−1+ · · · + Inj ⊗ A0 vec(Xλj) = 0, that is, A`XλjJ ` λ,j + A`−1XλjJ `−1 λ,j + · · · + A0Xλj = 0. (2.9)
It is convenient for us to write this canonical form in terms of pairs of matrix (Xλ, Jλ) where Xλ =Xλ1 · · · Xλq ∈ Cn×mλ (2.10a) and Jλ = diag(Jλ,1, . . . , Jλ,q) ∈ Cmλ×mλ, (2.10b)
The pair (Xλ, Jλ) is called a Jordan pair of P (λ) corresponding to eigenvalue λ.
Notice that from (2.9) and (2.10) it is easy to verify that
A`XλJλ`+ A`−1XλJλ`−1+ · · · + A0Xλ = 0. (2.11) Let Knj = 0 · · · 1 .. . ... 1 · · · 0 ∈ R nj×nj, j = 1, . . . , q, and K = diag(Kn1, · · · , Knq) ∈ R
mλ×mλ. It is easy to check that
KJλ>K = Jλ.
Similarly, we can construct a Jordan pair (YλK, Jλ) for the transposed matrix
polynomial P>(λ) corresponding to eigenvalue λ, where Yλ is complex conjugate
of complex matrix Yλ. From (2.11), we see that
A>` YλKJλ` + A >
`−1YλKJλ`−1+ · · · + A >
which is equivalent to
Jλ`YλHA`+ Jλ`−1Y H
λ A`−1+ · · · + YλHA0 = 0. (2.12)
We write Yλ in partitioned form
Yλ =Yλ1| · · · |Y q λ ∈ C
n×mλ (2.13)
which conforms with Jλ. The triple of matrices (Xλ, Jλ, Yλ) is called a Jordan triple
of P (λ) corresponding to λ.
Let λ1, . . . , λν be the distinct eigenvalues of PEP (1.2). A triple of matrices
(X, J, Y ), where X, YH are n × n` and J is a n` × n` Jordan matrix is called a Jordan triple of matrix polynomial P (λ) if the structure of X, Y and J is as follows X = [Xλ1, · · · , Xλν], Y = [Yλ1, · · · , Yλν] and J = diag(Jλ1, . . . , Jλν), (2.14)
where (Xλi, Jλi, Yλi), i = 1, . . . , ν are Jordan triples of P (λ).
In this paper, we shall consider in how the multiple eigenvalues change of the PEP (1.2) when the coefficient matrices, A0, . . . , A`, of P (λ) are subjected to small
perturbations. The usual formulation of the problem introduces a perturbation parameter ε, belong to some neighborhood of zero, and writes the perturbed matrix polynomial as P (λ) + εPB(λ) where
PB(λ) = λ`B`+ . . . + λB1+ B0 with kBik ≤ ωi for i = 0, 1, . . . , `. (2.15)
The nonnegative real numbers ωi, i = 0, 1, . . . , `, are the weights that represent
tolerances against which the perturbations are measure.
2.2
Preliminary results
The local Smith form of a matrix polynomial is described as follows.
Theorem 2.1. [1] Let P (λ) be an n × n matrix polynomial with det(P (λ)) ≡/ 0. Then for every eigenvalue λ1 ∈ C, P (λ) admits the representation
P (λ) = Eλ1(λ)diag (In−q, (λ − λ1)
nq, . . . , (λ − λ
1)n1) Fλ1(λ), (2.16)
where Eλ1(λ) and Fλ1(λ) are matrix polynomials invertible at λ1, and n1 ≥ n2 ≥
· · · ≥ nq ≥ 1 are the partial multiplicities of P (λ) corresponding to λ1.
Let λ1 ∈ C be a multiple eigenvalue of P (λ) with multiplicity mλ1 and let
n1, n2, . . . , nq be the corresponding partial multiplicities of P (λ) which satisfy the
order relation (2.7). In [2], the authors show that if P (λ) = λI + A then the splitting of λ1 under a small perturbation of A of order ε is, generally, of order
ε1/n1. In the next theorem, we show that if P (λ) is a regular matrix polynomial,
then the splitting of λ1 under a small perturbation of constant coefficient matrix
Theorem 2.2. Let λ1 ∈ C be a multiple eigenvalue of PEP whose largest Jordan
block has size n1 and let ω be a any positive number. Then for any positive number
ε, sufficiently small, there is a matrix B0 with kB0k ≤ ω such that cε1/n1 = |˜λ − λ1|,
where ˜λ is a eigenvalue of eP (λ) := P (λ) + εB0 and c > 0 is independent of ε.
Proof. By local Smith Theorem, there exist two matrix polynomials Eλ1(λ) and
Fλ1(λ), which are invertible at λ1, such that (2.16) holds. Since Eλ1(λ1) and Fλ1(λ1)
are invertible, there exists a positive number δ such that Eλ1(λ) and Fλ1(λ) are
invertible for all λ ∈ Dδ(λ1) ≡ {λ ∈ C||λ − λ1| ≤ δ}. Since Dδ(λ1) is a compact
set and kEλ1(λ)k, kFλ1(λ)k are continuous, by Extreme Value Theorem we obtain
two positive numbers mE, mF such that kEλ1(λ)k ≤ mE, kFλ1(λ)k ≤ mF for all λ
in Dδ(λ1). Let c = ω mEmF 1/n1 , (2.17)
then c > 0 is independent of ε. For any positive number ε with ε < (δc)n1, set
˜ λ = λ1+ cε1/n1 and B0 = Eλ1(˜λ) 0(n−1)×(n−1) 0(n−1)×1 01×(n−1) −cn1 Fλ1(˜λ). (2.18) Since |˜λ − λ1| = cε1/n1 < δ, we have kEλ1(˜λ)k ≤ mE and kFλ1(˜λ)k ≤ mF. (2.19) From (2.17)-(2.19), we obtain kB0k ≤ kEλ1(˜λ)kc n1kF λ1(˜λ)k ≤ mEmF ω mEmF = ω. (2.20)
From (2.16) and (2.18), we have
e P (˜λ) = P (˜λ) + εB0 = Eλ1(˜λ)diag In−q, (cε 1/n1)nq, . . . , (cε1/n1)n1 − cn1ε F λ1(˜λ) = Eλ1(˜λ)diag In−q, (cε 1/n1)nq, . . . , (cε1/n1)n2, 0 F λ1(˜λ).
Then eP (˜λ) is singular. Hence, ˜λ = λ1+ cε1/n1 is a eigenvalue of eP (λ) and |˜λ − λ1| =
Given a matrix polynomial P (λ) in (1.1). It is natural to consider that the tolerances of perturbation are made in Ai are equal to kAik, i.e., ωi = kAik for
i = 0, 1, . . . , `.
Corollary 2.3. Let λ1 ∈ C be a multiple eigenvalue of P (λ) whose largest Jordan
block has size n1 and let A0 be the coefficient matrix of P (λ). If either (i) A0 6= 0
or (ii) λ1 6= 0 holds, then there exists a matrix polynomial PB(λ) in (2.15) with
ωi = kAik such that the splitting of λ1 under a small perturbed P (λ) + εPB(λ) is
of order ε1/n1.
Proof. (i) Let ω = kA0k > 0. Theorem 2.2 shows that there exists a matrix B0
with kB0k ≤ ω = kA0k, let PB(λ) = B0, then the splitting of λ1 under a small
perturbed P (λ) + εPB(λ) is of order ε1/n1.
(ii) We need only consider the case A0 = 0. Since P (λ) is regular, there is a
positive integer k, 0 < k < `, such that
P (λ) = λkQ(λ)
= λk(λ`−kA`+ λ`−k−1A`−1+ · · · + Ak)
with Ak 6= 0. It is easy to see that λ1 is a eigenvalue of Q(λ) because λ1 6= 0. Let
ω = kAkk > 0. From Theorem 2.2 we have a matrix B0 with kB0k ≤ ω = kAkk
such that the splitting of λ1 under a small perturbed Q(λ) + εB0 is of order ε1/n1.
Let PB(λ) = λkB0, then it is easy to check that the splitting of λ1 under a small
perturbed P (λ) + εPB(λ) is of order ε1/n1.
It is natural to ask if A0 = 0 and λ1 = 0 then can we find a matrix polynomial
PB(λ) in (2.15) with ωi = kAik such that the first-order term of the expansion of
eigenvalue of P (λ) + εPB(λ) is of order ε1/n1?. The answer is negative, we give a
counterexample. Let P (λ) = λI (in our notation, ` = 2, A1 = I and A0 = 0).
Then λ1 = 0 is the only eigenvalue of P (λ) whose largest Jordan block has size
n1 = 1. For arbitrary matrix polynomial PB(λ) = λB1+ B0 with kB1k ≤ kA1k and
kB0k ≤ kA0k, it follows that PB(λ) = λB1 with kB1k ≤ 1. It is easy to check that
λ(ε) = 0 is the only eigenvalue of P (λ) + εPB(λ) = λ(I + εB1). So, the splitting of
λ1 = 0 under a small perturbed P (λ) + εPB(λ) is not of order ε1.
Given a regular matrix polynomial P (λ) in (1.1). Suppose λ1 ∈ C is a multiple
eigenvalue of P (λ) and the dimension of largest Jordan block is n1. We define a set
Pεthat collects all perturbed matrix polynomial which we allow under perturbation
parameter ε : Pε = n e P (λ) = P (λ) + εPB(λ) : PB(λ) forms (2.15) with kBik ≤ kAik, i = 0, 1, . . . , ` . (2.21)
Let (Jλ1, Xλ1) be a Jordan pair of P (λ). For each eP (λ) ∈ Pε, suppose that eJλ1 = Jλ1+ ∆Λ and eXλ1 = Xλ1 + ∆X satisfy (A`+ εB`) eXλ1Je ` λ1 + · · · + (A1+ εB1) eXλ1Jeλ1 + (A0+ εB0) eX = 0. (2.22) where ∆Λ = diag(δ1, δ2, . . . , δmλ1) ∈ Cmλ1×mλ1, ∆X ∈ Cn×mλ1. (2.23)
From (2.22), we have λ1 + δ1, λ1 + δ2, . . . , λ1 + δmλ1 are eigenvalues of eP (λ). By
Corollary 2.3, we obtain that if either A0 6= 0 or λ1 6= 0 holds, then
δε := max e P ∈Pε
k∆Λk = O(ε1/n1). (2.24)
Remark 2.1. Let P (λ) be a matrix polynomial and let λ1 ∈ C be a multiple
eigenvalue of P (λ) whose largest Jordan block has size n1. If either A0 6= 0 or
λ1 6= 0, then
lim sup
ε→0
δε
ε1/n1 = κ (2.25)
where δε is given by (2.24) and κ is positive real number.
We have discussed the order of splitting of multiple eigenvalues under the per-turbed matrix polynomial eP (λ). Now we focus on the order of splitting of eigen-vector matrix X.
Lemma 2.1. Let A be a square complex singular matrix and b be a complex vector with kbk = 1. If Ax = εb is consistent then the minimal solution xb is of order ε.
Proof. Let A have the Singular value decomposition
A = U Σ 0 0 0 V ≡ [U1, U2] Σ 0 0 0 V (2.26)
where U , V are unitary and Σ ia diagonal. Since Ax = b is consistent and
UHAV VHx = Σ 0 0 0 (VHx) = ε U H 1 b U2Hb . (2.27) Thus UH
2 b = 0 and kU1Hbk = kbk = 1. From (2.27) the minimal solution is
xb = εV Σ−1UH 1 b 0 . (2.28) Hence kxbk = εkΣ−1kkbk = O(ε).
Theorem 2.4. Let λ1 ∈ C be a multiple eigenvalue of P (λ) with multiplicity m1
and let λ1 + δi, i = 1, . . . , mλ1, are the eigenvalues of eP (λ) ∈ Pε. If λi + δi,
i = 1, . . . , mλ are eigenvalues of Pε and
e
P (λ) ∈ Pε. If ∆Λ = diag(δ1, δ2, . . . , δm) with k∆Λk = O(εα), α ≤ 1, such
that λ1 + δi, i = 1, . . . , mλ1, are eigenvalues of eP (λ) then there exists ∆X with
k∆Xk ≤ cεα such that (2.22) holds where c is a nonnegative number.
Proof. Let eP (λ) ∈ Pε and eJ = J + ∆Λ be the eigenvalue matrix of eP (λ) then there
exists an n × m matrix ∆X such that (2.22) holds. Since α ≤ 1
A`XJ`+ · · · + A1XJ + A0X = 0, we have A`∆XJ`+ A`−1∆XJ`−1+ · · · + A0∆X =A`X(∆ΛJ`−1+ J ∆ΛJ`−2+ · · · + J`−1∆Λ) + A`−1X(∆ΛJ`−2+ J ∆ΛJ`−3 + · · · + J`−2∆Λ) + · · · + A1X∆Λ + ε(B`XJ`+ B`−1XJ`−1+ · · · + B0X) + o(εα) ≤ξεα, (2.29)
where ξ is a nonnegative number. Take vec operation of (2.29) then we obtain
J`>⊗ A`+ J(`−1)>⊗ A`−1+ · · · + Im⊗ A0 vec(∆X) = b, (2.30)
where b ∈ Cnm and kbk ≤ ξεα. Since linear system (2.30) is consistent, by Lemma 2.1 we obtain that there exists an n × m complex matrix ∆X such that vec(∆X) is the solution of (2.30) and kvec(∆X)k ≤ cεα for some nonnegative c.
Thus k∆Xk ≤ cεα.
3
Sensitivity of generalized eigenvalue problem
In this section, we consider a generalized eigenvalue problem
Ax = λBx,
where G(λ) ≡ λB − A is regular matrix polynomial of degree 2, λ ∈ C and x ∈ Cn are eigenvalue and eigenvector of G(λ). Let λ = λ1 be a finite eigenvalue of G(λ)
with multiplicity mλ. Suppose that n1, n2, . . . , nqare the partial multiplicities for λ
which satisfy the order relation (2.7). Let ε be a perturbation parameter, belonging to some neighborhood of zero. We write the perturbed matrix pencil
for some arbitrary matrices E and F . Suppose λ(ε) with λ(0) = λ1 is a solution
curve of the characteristic equation
det (G(λ(ε), ε)) = 0.
Theorem 2.2 shows that there exists a perturbation matrix E and an eigenvalue curve λ(ε) of (3.31) with F = 0, such that λ(ε) admits a first-order expansion
λ(ε) = λ1+ cε1/n1+ o(ε1/n1) (3.32)
where c is a nonzero constant. In this section, we want to find the leading coefficient c of the eigenvalue curve λ(ε) in (3.31).
Let (Yλ, Jλ, Xλ) be the Jordan triples of G(λ) corresponding to eigenvalue λ =
λ1 and let
Aλ = YλHAXλ, Bλ = YλHBXλ, Eλ = YλHEXλ, Fλ = YλHF Xλ. (3.33)
The following two lemmas are useful for the finding the leading coefficient of λ(ε).
Lemma 3.1. [3, p.306] Let (Yλ, Jλ, Xλ) be the Jordan triples of G(λ) corresponding
to eigenvalue λ = λ1. Then there are nonsingular matrices X = [Xλ, X2] and
Y = [Yλ, Y2] such that YH λ YH 2 A[Xλ, X2] = Aλ 0 0 A2 and Y H λ YH 2 B[Xλ, X2] = Bλ 0 0 B2 ,
where Aλ, Bλ are given in (3.33) and A2 = Y2HAX2, B2 = Y2HBX2. Furthermore,
Aλ = BλJλ, Aλ = JλBλ and Bλ ∈ Cmλ×mλ is invertible.
Lemma 3.2. [3, p.310] Let X = [Xλ, X2] and Y = [Yλ, Y2] be defined as Lemma 3.1.
Given the perturbation matrices E and F , let
YH λ Y2H E[Xλ, X2] = Eλ E12 E21 E22 and Y H λ Y2H F [Xλ, X2] = Fλ F12 F21 F22 .
If the perturbation parameter ε is sufficiently small, then the eigenvalues of per-turbed pencil G(λ, ε) in (3.31) are the eigenvalues of
e
Gλ(λ, ε) = λ(Bλ+ εFλ+ εF12P ) − (Aλ+ εEλ+ εE12P ) (3.34)
and
e
G2(λ, ε) = λ(B2+ εF22+ εQF12) − (A2+ εE22+ εQE12)
From Lemma 3.1, we have
JλBλ = Aλ = BλJλ,
where Aλ, Bλ, Jλ ∈ Cmλ×mλ and Bλ is invertible. This implies that λ1 is only
eigenvalue of λBλ − Aλ with multiplicity mλ and that Bλ commutes with Jλ. Let
Xλ, Yλ and Jλ in the form of (2.13), (2.10a) and (2.10b), respectively. If we write
Bλ in partitioned form Bλ = (Bij), where Bij = Yi
H λ BX j λ ∈ Cni ×ni, and Yi λ and X j λ
are the ith and jth column block of Yλand Xλ, respectively. An explicit calculation
shows that each Bij ∈ Cni×ni (note that n1 ≥ n2 ≥ · · · ≥ nq) is of the form
Bij = [0, bBij] if i ≥ j, b Bij 0 if i ≤ j, (3.35)
and bBij ∈ Cs×s (s = min{ni, nj}) is an upper triangular matrix of Toeplitz type,
that is, b Bij =
b(ij)1 b(ij)2 · · · b(ij)s
0 b(ij)1 . .. ... .. . . .. ... b(ij)2 0 · · · 0 b(ij)1 (3.36)
where b(ij)1 , . . . , b(ij)s are constants. In the case ni ≥ nj, it is easy to see that
[b(ij)1 , b(ij)2 , · · · , b(ij)s ] = yiHBXλj (3.37) where yi is the first column of Yλi.
Note that from Lemma 3.2, we conclude that to investigate the eigenvalue curves λ(ε) with λ(0) = λ1, we only need to consider the perturbed pencil (3.34) for ε is
sufficiently small. Since n1 is the largest dimension of Jordan blocks corresponding
to eigenvalue λ. Theorem 2.2 says that there exists a perturbed pencil such that the perturbed pencil has an eigenvalue curve formed with (3.32). Since we are interested in the leading coefficient c (this is the coefficient of ε1/n1) in (3.32) and
the matrix norm of εF12P and εE12P are of order ε2, we can ignore those two
terms in the perturbed matrix pencil (3.34). Then we consider the perturbed matrix pencil
Gλ(ω, ε) = ω(Bλ+ εFλ) − (Aλ+ εEλ). (3.38)
Suppose ω(ε) with ω(0) = λ1 is an eigenvalue curve of (3.38), then we have
We perform on (3.38) the change of variables z = ε1/n1, µ = ω − λ1 z . (3.40) This leads to b Gλ(µ, z) = (λ1 + zµ)(Bλ+ zn1Fλ) − (Aλ + zn1Eλ),
where λ1 is only eigenvalue of Gλ(ω, 0). Using Aλ = BλJλ and (2.7), it follows that
b Gλ(µ, z) = Bλ(λ1+ zµ − Jλ) + zn1((λ1+ zµ)Fλ− Eλ) = Bλ(zµ − Nλ) + zn1((λ1+ zµ)Fλ − Eλ), (3.41) where Nλ = diag(Ir1 ⊗ Nnr1, . . . , Irp⊗ Nnrp) (3.42) and Nnri = 0 1 0 . .. . .. 1 0 ∈ Rnri×nri, for i = 1, 2, . . . , p. Denote Lλ(z) = diag(L1(z), . . . , Lq(z)), Rλ(z) = diag(R1(z), . . . , Rq(z)), where Li(z) = diag(z−1, z−2, . . . , z−ni), Ri(z) = diag(1, z1, . . . , zni−1)
for i = 1, 2, . . . , q. We now introduce the matrix
ˇ Gλ(µ, z) = L(z) bGλ(µ, z)R(z) = L(z)Bλ(zµ − Nλ) R(z) + zn1L(z) ((λ1+ zµ)Fλ− Eλ) R(z) ≡ ˇG1(µ, z) + ˇG2(µ, z), (3.43) and define Q(µ, z) = det ˇGλ(µ, z) . (3.44)
Since L(z) and R(z) are invertible for any z 6= 0, we conclude that det( bGλ(µ, z)) = 0 ⇔ Q(µ, z) = 0, for any z 6= 0.
We are mainly concerned with the solutions of Q(µ, z) = 0 when z is close to zero. First, we analysis the matrix ˇG1(µ, z) given in (3.43). To see this, let us
consider
Nλ(z) = L(z)NλR(z),
Bλ(z) = L(z)BλL−1(z),
(3.45)
where the matrices Nλ and Bλ are defined in (3.42) and (3.33), respectively. A
simple calculation shows that L(z)NλR(z) = Nλ which is independent of z. Hence,
the matrix N (z) is a constant matrix. Now we consider the matrix Bλ(z) in (3.45).
Since Bλ = (Bij) and each submatrix Bij ∈ Cni×nj is of the form (3.36), which is
upper triangular, then we obtain that
Bij(z) = Li(z)BijL−1j (z) ∈ C ni×nj
is matrix polynomial in z and Bλ(z) = (Bij(z)). From (3.35) and (3.36), it is easy
to check that Bij(0) = 0 if ni < nj, b(ij)1 Inj 0 if ni ≥ nj. (3.46)
Since the partial multiplicities ni, i = 1, 2, . . . , q, satisfy the order relation (2.7),
from (3.46) we have that the matrix Bλ(0) is a block upper triangular matrix. Now
we focus on the diagonal blocks of Bλ(0). Based on (3.37), let xi and yi be the first
columns of Xi
λ and Yλi, respectively, then we have
b(ij)1 = yiHBxj, if ni ≥ nj.
We build up two matrices
Xλ,1i = [xri−1+1, . . . , xri], Y
i
λ,1 = [yri−1+1, . . . , yri], (3.47)
where ri, i = 1, 2, . . . , p, are given as in (2.7) and denote
Ti = Yi
H
λ,1BX i
λ,1, for i = 1, 2, . . . , p. (3.48)
From (3.46), the structure of Bλ(0) must be of the form
Bλ(0) = T1 ⊗ Inr1 ? . .. 0 Tp⊗ Inrp ∈ C mλ×mλ. (3.49)
We known from Lemma 3.1 that Bλ is invertible. It is easy to verify that Bλ(0) is
Lemma 3.3. Given ˇG1(µ, z) = L(z)Bλ(zµ − Nλ)R(z) in (3.43). Then
(i) each entry of ˇG1(µ, z) is a polynomial in z.
(ii) when z = 0, we have
ˇ
G1(µ, 0) = Bλ(0)(µImλ− Nλ) (3.50)
where Bλ(0) in (3.49) is an invertible matrix.
Proof. Since L(z)(zµImλ)R(z) = µImλ. Then ˇ G1(µ, z) = L(z)Bλ(zµ − Nλ)R(z) = L(z)BλL−1(z)(L(z)zµImλR(z) − L(z)NλR(z)) = Bλ(z)(µImλ− Nλ(z)),
where N (z) and bB(z) are defined in (3.45). Using the fact that bB(z) is matrix polynomial in z and that Nλ(z) = Nλ is independent of z then we obtain that each
entry of
ˇ
G1(µ, z) = Bλ(z)(µIm1 − Nλ)
is a polynomial in z. The second statement is straight forward.
Next, we analysis the matrix ˇG2(µ, z) = zn1L(z) ((λ1+ zµ)Fλ− Eλ) R(z) where
Fλ and Eλ are given as in (3.33). We write Fλ and Eλ in partitioned form
Fλ = (Fij) and Eλ = (Eij), where Fij = Yi H λ F X j λ ∈ C ni×nj and F ij = Yi H λ EX j λ ∈ C ni×nj. Let ˇ L(z) ≡ diag( ˇL1(z), . . . , ˇLq(z)) = zn1L(z)ˇ and Fλ(z) = ˇL(z)FλR(z) = (Fij(z)), Eλ(z) = ˇL(z)EλR(z) = (Eij(z)), (3.51)
where ˇLi(z) = diag(zn1−1, . . . , zn1−ni), Fij(z) = ˇLi(z)FijRj(z) and Eij(z) = ˇLi(z)EijRj(z)
n1 ≥ ni. From (3.51), it is easy to see that Fλ(z) and Eλ(z) are matrix polynomial in z. Taking z = 0 in (3.33) we obtain Fij(0) = 0 0 fij 0 1 ≤ i ≤ r1, 0 otherwise, and Eij(0) = 0 0 eij 0 1 ≤ i ≤ r1, 0 otherwise,
where fij and eij are the entries in the nith row and first column of Fij and Eij,
respectively. Since Fij(z) = ˇLi(z)FijRj(z) and Eij(z) = ˇLi(z)EijRj(z), we have
fij = yi H niF x j 1 and eij = yi H niEx j 1,
where yini is the nith column of Yλi and x j
1 is the first column of X j λ. We build a matrix Yλ,n1 i = [yn11, yn22, . . . , yr1 nr1] ∈ C n×r1. (3.52)
The structure of Fλ(0) and Eλ(0) must be of the form
Fλ(0) = Y1H λ,niF X 1 λ,1⊗ Υn1 ? 0 0 and Eλ(0) = Y1H λ,niEX 1 λ,1 ⊗ Υn1 ? 0 0 , (3.53) where Y1H λ,niF X 1 λ,1, Y1 H λ,niEX 1 λ,1 ∈ Cr1 ×r1, X1 λ,1 is given as in (3.47) and Υn1 = 0 0 · · · 0 .. . ... ... 0 0 · · · 0 1 0 · · · 0 ∈ Rn1×n1.
Then we have following results.
Lemma 3.4. Given ˇG2(µ, z) = zn1L(z) ((λ1+ zµ)Fλ− Eλ) R(z) in (3.43). Then
(i) each entry of ˇG2(ν, z) is a polynomial in z.
(ii) when z = 0, we have ˇ
G2(µ, 0) = λ1Fλ(0) − Eλ(0), (3.54)
Substituting (3.50) and (3.54) into (3.44) with z = 0, we have
Q(µ, 0) = det (Bλ(0)(µImλ− Nλ) + λ1Fλ(0) − Eλ(0)) .
Using the structures of Bλ(0), Nλ, Fλ(0) and Eλ(0) which are given as in (3.49),
(3.42) and (3.52). we obtain Q(µ, 0) = c1µmλ−r1n1det T1⊗ (µIn1 − Nn1) + Y 1H λ,ni(λ1F − E)X 1 λ,1⊗ Υn1 , (3.55)
where c1 = det(T2⊗Inr2) · · · det(Tp⊗Inrp) and Ti, i = 1, 2, , p, are given as in (3.48).
Lemma 3.1 shows that the matrix Bλ(0) in (3.49) is invertible, hence c1 6= 0.
Lemma 3.5. Let W = Yλ,n1H i(λ1F − E)X 1 λ,1 ∈ C r1×r1. (3.56)
The solution set of Q(µ, 0) = 0 in (3.55) is
Ψ = {µ | det (µn1T
1+ W ) = 0} ∪ {0} (3.57)
where T1 = Y1
H
λ,1BXλ,11 and Xλ,11 , Yλ,11 and Yλ,n1 i are given in (3.47) and (3.52).
Proof. From the definition of Kronecker product, it is easy to check
Q(µ, 0) = cµm1−r1n1det ((µI n1 − Nn1) ⊗ T1+ Υn1 ⊗ W ) . Let U = 0 · · · 0 1 1 . .. µ . .. ... 1 µn1−1 ∈ Rn1×n1,
then det(I ⊗ U )= det(U ) = 1. Compute the two matrices
(µIn1 − Nn1)U = −1 µ −1 . .. ... µ µn1 , Υn1U = 0 · · · 0 0 .. . ... ... 0 · · · 0 0 0 · · · 0 1 ,
then it is easy to verify that the matrix
is block lower triangular matrix and
det [((µIn1 − Nn1) ⊗ T1+ Υn1 ⊗ W )]
= det [((µIn1 − Nn1) ⊗ T1+ Υn1 ⊗ W ) U ⊗ Ir1]
= det(−T1)n1−1det (µn1T1+ W )
Since T1 is invertible, we have det(−T1)n1−1 6= 0. Thus the root set of Q(µ, 0) = 0
is {µ | det (µn1T
1+ W ) = 0} ∪ {0}.
The following theorem is our main result.
Theorem 3.1. Let A − λB be a regular pencil. Suppose that λ1 is a multiple
eigenvalue of A − λB whose largest Jordan block has size n1 and r1 is the number of
Jordan blocks with dimension n1. Then there are r1n1 eigenvalues of the perturbed
matrix pencil (A + εE) + λ(B + εF ) admitting a first-order expansion
λki(ε) = λ1+ (ξk)1/n1ε1/n1 + o(ε1/n1) (3.58)
for k = 1, . . . , r1, i = 1, . . . , n1, where ξk, k = 1, . . . , r1, are the roots of
det (ξT1+ W ) = 0, (3.59)
where T1 and W are given in (3.48) and (3.56), respectively.
Proof. From (3.43), we know that for any fixed z with z 6= 0, the solution sets of Q(µ, z) ≡ det( ˇGλ(µ, z)) = 0 and det( bGλ(µ, z)) = 0 are equal. Lemma 3.3 and
Lemma 3.4 show that each entry of ˇGλ(µ, z) = ˇG1(µ, z) + ˇG2(µ, z) is a polynomial
in z. From Lemma 3.5, we conclude that
n
µ| det( bGλ(µ, z)) = 0
o
→ Ψ, as z → 0.
Using the fact that Ψ in (3.57) has at most r1n1 nonzero elements and from
(3.40), we obtain that there are r1n1 eigenvalue curves of (3.38) ωik(ε) admit the
first-order expansion
ωki(ε) = λ1+ µε1/n1 + o(ε1/n1)
= λ1+ (ξk)1/n1ε1/n1 + o(ε1/n1)
for k = 1, . . . , r1, i = 1, . . . , n1, where ξ1, . . . , ξr1 are the roots of det (ξT1+ W ) = 0
and T1, W are given in (3.48), (3.56), respectively.
The equation (3.39) shows that ωik(ε) and the eigenvalue curves λki(ε) with λk
i(0) = λ1 of (A + εE) + λ(B + εF ) have the same first-order expansion. Hence,
Acknowledgments
This work is partially supported by The National Center of Theoretical Sciences, of R.O.C. on Taiwan.
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