Table Of ContentVector space of linearizations for the quadratic
two-parameter matrix polynomial
Bibhas Adhikari∗
2
1 Indian Institute of Technology Rajasthan
0
2 Jodhpur, India
n
a
J
6 Abstract: Given a quadratic two-parameter matrix polynomial Q(λ,µ), we de-
velopasystematicapproachtogeneratingavectorspaceoflineartwo-parameter
]
A matrix polynomials. We identify a set of linearizations of Q(λ,µ) that lie in the
N vector space. Finally, we determine a class of linearizations for a quadratic two-
. parameter eigenvalue problem.
h
t
a
m Key words: matrix polynomial, two-parameter matrix polynomial, quadratic
[ two-parametereigenvalueproblem, two-parametereigenvalueproblem, lineariza-
1 tion
v
0
AMS classification: 65F15, 15A18, 15A69, 15A22
5
3
1
. 1 Introduction
1
0
2
1 We consider two-parameter quadratic matrix polynomials of the form
:
v
i Q(λ,µ) = A+λB +µC +λ2D +λµE +µ2F (1)
X
r
a where λ,µ are scalars and the coefficient matrices are real or complex matrices
of order n×n. If (λ,µ) ∈ C×C and nonzero x ∈ Cn satisfy Q(λ,µ)x = 0, then
x is said to be an eigenvector of Q(λ,µ) corresponding to the eigenvalue (λ,µ).
The classical approach to solving spectral problems for matrix polynomials is
to first perform a linearization, that is, to transform the given polynomial into
a linear matrix polynomial, and then work with this linear polynomial (see
[16, 19, 17, 24, 23, 21] and the references therein). Therefore, given a quadratic
two-parameter matrix polynomial Q(λ,µ), we seek linear two-parameter matrix
polynomials L(λ,µ) = λA + µA + A , called linearizations, which have the
1 2 3
same eigenvalues as Q(λ,µ).
b b b
∗
E-mail: [email protected]
The one-parameter matrix polynomials have been an active area of research
in numerical linear algebra [21, 27, 28]. In [21], Mackey et al. have investi-
gated the one-parameter polynomial eigenvalue problem extensively and they
have produced vector spaces of linearizations for a one-parameter matrix poly-
nomial by generalizing the companion forms of the one-parameter polynomial.
Adopting a similar approach we derive a set of linearizations of a quadratic
two-parameter matrix polynomial.
The quadratic two-parameter eigenvalue problem is concerned with finding
scalars λ,µ ∈ C and non-zero vectors x ∈ Cn1,x ∈ Cn2 such that
1 2
Q (λ,µ)x = (A +λB +µC +λ2D +λµE +µ2F )x = 0
1 1 1 1 1 1 1 1 1 (2)
Q (λ,µ)x = (A +λB +µC +λ2D +λµE +µ2F )x = 0
2 2 2 2 2 2 2 2 2
(cid:26)
where A ,B ,...,F ∈ Cni×ni,i = 1,2. A pair (λ,µ) satisfying (2) is called
i i i
an eigenvalue of (2) and x ⊗ x , where ⊗ is the Kronecker product, is the
1 2
corresponding eigenvector. This problem appears in stability analysis of dif-
ferent systems, for example, time-delay systems of single delay [9, 10, 13, 24].
The standard approach to solving (2) is by linearizing the problem into a two-
parameter eigenvalue problem of larger size and then by converting it into an
equivalent coupled generalized eigenvalue problem which is then solved by nu-
merical methods, see [24, 23, 9].
Given (2), we seek a two-parameter eigenvalue problem
L (λ,µ)w := (A(1))+λB(1) +µC(1))w = 0
1 1 1 (3)
L (λ,µ)w := (A(2))+λB(2) +µC(2))w = 0
2 2 2
(cid:26)
with the same eigenvalues, where w ∈ C3ni \{0} and A(i),B(i),C(i) ∈ C3ni×3ni,
i
i = 1,2. In such case (3) is called a linearization of (2).
The choice of a linearization may have an adverse effect on the sensitivity
of the eigenvalues. Therefore, it is important to identify potential lineariza-
tions and describe their constructions. In this paper, we develop a systematic
approach that enables us to generate a class of linearizations for a quadratic
two-parameter eigenvalue problem.
2 Linearizations for quadratic two-parameter
matrix polynomial
In this section we construct a set of linearizations of a quadratic two-parameter
matrix polynomial.
Definition 2.1 ([24]) A ln × ln linear matrix polynomial L(λ,µ) = λA +
1
µA + A is a linearization of an n × n matrix polynomial Q(λ,µ) if there
2 3
exist polynomials P(λ,µ) and R(λ,µ), whose determinant is a non-zero consbtant
inbdependbent of λ and µ, such that
Q(λ,µ) 0
= P(λ,µ)L(λ,µ)R(λ,µ).
0 I(l−1)n
(cid:20) (cid:21)
2
Let Q(λ,µ) be a quadratic two-parameter matrix polynomial given by
Q(λ,µ) = λ2A +µ2A +λµA +λA +µA +A
20 02 11 10 01 00
wherethecoefficientmatricesareofordern×n.Assumethatxistheeigenvector
corresponding to an eigenvalue (λ,µ) of Q(λ,µ), that is, Q(λ,µ)x = 0. With
a view to constructing linearizations of Q(λ,µ), we denote x = x ,λx =
00 00
x ,µx = x . Then we have
10 00 01
A (λx )+A (µx )+A (λx )+A x +A x +A x = 0. (4)
20 10 02 01 11 01 10 10 01 01 00 00
Consequently we have
A A 0 0 A 0 A A A x
20 11 02 10 01 00 10
λ 0 0 0 +µ 0 0 I + 0 −I 0 x = 0. (5)
01
0 0 I 0 0 0 −I 0 0 x
00
b b b
A1 A2 A3
| {z } | {z } | {z }
x λx λ λ
10
Observe that x = µx = µ ⊗ x. We denote Λ := µ . Thus x is
01
x x 1 1
00
the eigenvector corresponding to an eigenvalue (λ,µ) of Q(λ,µ) if and only if
L(λ,µ)w = 0 where w = Λ ⊗ x and L(λ,µ) = λA + µA + A , that is, w is
1 2 3
the eigenvector corresponding to an eigenvalue (λ,µ) of L(λ,µ). We show that
L(λ,µ) is a linearization of Q(λ,µ). b b b
Define
λI I 0
n n
E(λ,µ) := µI 0 I ,
n n
I 0 0
n
I µA +λA +A λA +A
n 02 11 01 20 10
F(λ,µ) := 0 0 I .
n
0 I 0
n
Notice that E,F are unimodular quadratic two-parameter matrix polynomials,
that is, determinants of E and F are constants. Then we can easily check that
Q(λ,µ) 0
F(λ,µ)L(λ,µ)E(λ,µ) = . (6)
0 I
2n
(cid:20) (cid:21)
Thus we have detQ(λ,µ) = γdetL(λ,µ) for some γ 6= 0. This implies L(λ,µ)
preserves the eigenvalues of Q(λ,µ) andhence is a linearization of order 3n×3n.
We call this linearization the standard linearization of Q(λ,µ). It is interesting
to observe that the linearization proposed in [24] is up to some permutations of
block rows and columns of the standard linearization.
3
However, for the standard linearization we have
L(λ,µ)·(Λ⊗x) = (Q(λ,µ)x)T 0 ... 0 T for all x ∈ Cn, (7)
and therefore, any solution o(cid:2)f (5) gives a soluti(cid:3)on of the original problem
Q(λ,µ)x = 0. Further, by (7) we have
λI Q(λ,µ) 1
n
L(λ,µ)·(Λ⊗I ) = L (λ,µ) µI = 0 = e ⊗Q(λ,µ),e = 0 . (8)
n 1 n 1 1
I 0 0
n
We restrict our attention to the equation (8) which is satisfied by the stan-
dard linearization. It would be worthy to find linear two-parameter matrix
polynomials L(λ,µ) that satisfy
L(λ,µ)·(Λ⊗I ) = v⊗Q(λ,µ) (9)
n
for some vector v ∈ C3. Therefore, we introduce the notation
V = {v⊗Q(λ,µ) : v ∈ K3} (10)
Q
and define
L(Q(λ,µ)) := L(λ,µ) = λA +µA +A ,A ∈ K3n×3n : L(λ,µ)·(Λ⊗I ) ∈ V .
1 2 3 i n Q
n (11) o
Note that L(Q(λ,µ)) 6= ∅ asbthe stabndardblinbearization L(λ,µ) ∈ L(Q(λ,µ)). It
is easy to check that L(Q(λ,µ)) is a vector space. If L(λ,µ) ∈ L(Q(λ,µ)) for
some v ∈ C3 then call v is an ansatz vector associated with L(λ,µ). To investi-
gate the structure of each L(λ,µ) ∈ L(Q(λ,µ)), we define a “box-addition” for
three 3n×3n block matrices as follows.
Definition 2.2 Let X,Y,Z ∈ C3n×3n be three block matrices of the form
X X X Y Y Y Z Z Z
11 12 b b13 b 11 12 13 11 12 13
X = X X X ,Y = Y Y Y ,Z = Z Z Z .
21 22 23 21 22 23 21 22 23
X X X Y Y Y Z Z Z
31 32 33 31 32 33 31 32 33
b b b
Define
X X X Y Y Y Z Z Z
11 12 13 11 12 13 11 12 13
X ⊞Y ⊞Z = X X X ⊞ Y Y Y ⊞ Z Z Z
21 22 23 21 22 23 21 22 23
X X X Y Y Y Z Z Z
31 32 33 31 32 33 31 32 33
b b b X X 0 X 0 0 0 Y Y 0 Y 0
11 12 13 11 12 13
= X X 0 X 0 0 + 0 Y Y 0 Y 0
21 22 23 21 22 23
X X 0 X 0 0 0 Y Y 0 Y 0
31 32 33 31 32 33
0 0 0 Z Z Z
11 12 13
+ 0 0 0 Z Z Z
21 22 23
0 0 0 Z Z Z
31 32 33
where ‘+’ is the usual matrix addition.
4
For the standard linearization L(λ,µ) = λA + µA + A ∈ L(Q(λ,µ)) we
1 2 3
have
b b b
A A 0 0 A 0 A A A
20 11 02 10 01 00
A ⊞A ⊞A = 0 0 0 ⊞ 0 0 I ⊞ 0 −I 0
1 2 3
0 0 I 0 0 0 −I 0 0
b b b A A 00 0 0 0 0 A 0 0 0
20 11 02
= 0 0 0 0 0 0 + 0 0 0 0 I 0
0 0 0 I 0 0 0 0 0 0 0 0
0 0 0 A A A
10 01 00
+ 0 0 0 0 −I 0
0 0 0 −I 0 0
A A A A A A
20 11 02 10 01 00
= 0 0 0 0 0 0
0 0 0 0 0 0
= e ⊗ A A A A A A .
1 20 11 02 10 01 00
Thus we have the following l(cid:2)emma. (cid:3)
Lemma 2.3 Let Q(λ,µ) = λ2A + µ2A + λµA + λA + µA + A be
20 02 11 10 01 00
a quadratic two-parameter matrix polynomial with real or complex coefficient
matrices of order n×n, and L(λ,µ) = λA +µA +A a 3n×3n two-parameter
1 2 3
linear matrix polynomial. Then
b b b
L(λ,µ)·(Λ⊗I ) = v ⊗Q(λ,µ) ⇔
n
A ⊞A ⊞A = v ⊗ A A A A A A .
1 2 3 20 11 02 10 01 00
(cid:2) (cid:3)
Proof: Computational abnd eabsy tobcheck.
Example 2.4 Consider a quadratic two-parameter polynomial
Q(λ,µ) = λ2A +µ2A +λµA +λA +µA +A
20 02 11 10 01 00
where A ∈ Cn×n and
ij
A A +A A +A
20 11 20 10 01
L(λ,µ) = λ A A 0
20 00
2A A +2A I
20 02 11
−A A A −A 0 A
20 02 01 01 00
+µ A −A A 0 + A A A .
11 00 02 01 01 00
−A 2A A −I +2A A 2A
02 02 01 10 01 00
Then L(λ,µ) ∈ L(Q(λ,µ)) since
A ⊞A ⊞A = [1 1 2]T ⊗ A A A A A A .
1 2 3 20 11 02 10 01 00
(cid:2) (cid:3)
b b b
5
Using Lemma 2.3 we characterize the structure of any L(λ,µ) ∈ L(Q(λ,µ)).
Theorem 2.5 Let Q(λ,µ) = λ2A + µ2A +λµA +λA + µA + A be
20 02 11 10 01 00
a quadratic two-parameter matrix polynomial with real or complex coefficient
matrices of order n × n, and v ∈ C3. Then a linear two-parameter matrix
polynomial L(λ,µ) ∈ L(Q(λ,µ)) corresponding to the ansatz vector v is of the
form L(λ,µ) = λA +µA +A where
1 2 3
A = v ⊗A −Y +v ⊗A −Z +v ⊗A
1 b b 20b 1 11 1 10
A2 = (cid:2)Y1 v ⊗A02 −Z2 +v ⊗A01 (cid:3)
b
A3 = (cid:2)Z1 Z2 v ⊗A00 (cid:3)
b
Y (cid:2) Z (cid:3)Z
b11 11 12
where Y = Y ,Z = Z ,Z = Z ∈ C3n×n are arbitrary.
1 21 1 21 2 22
Y Z Z
31 31 32
Proof: LetM : L(Q(λ,µ)) → V beamultiplicativemapdefinedbyL(λ,µ) 7→
Q
L(λ,µ)(Λ ⊗ I ). Its easy to see that M is linear. First we show that M is
n
surjective. Let v ⊗Q(λ,µ) be an arbitrary element of V . Construct L(λ,µ) =
Q
λA +µA +A where
1 2 3
A = v ⊗A v ⊗A v ⊗A
b b b 1 20 11 10
A2 = (cid:2)0 v ⊗A02 v⊗A01 (cid:3)
b
A3 = (cid:2)0 0 v ⊗A00 . (cid:3)
b
Then obviously we habve A1⊞(cid:2)A2⊞A3 = v⊗(cid:3) A20 A02 A11 A10 A01 A00 ,
so by Lemma 2.3 L(λ,µ) is an M-pre-image of v ⊗Q(λ,µ). The set of all M-
(cid:2) (cid:3)
preimages of v ⊗Q(λ,µ)bis L(λb,µ)+bKerM, so all that remains is to compute
KerM. Further by Lemma 2.3 KerM contains L(λ,µ) = λA + µA + A
1 2 3
that satisfies A ⊞A ⊞A = 0. The definition of “box-addition” implies that
1 2 3
A ,A ,A are of the following form b b b
1 2 3
b b b
A = 0 −Y −Z
b b b 1 1 1
A2 = (cid:2)Y1 0 0 (cid:3)
b
A3 = (cid:2)Z1 Z2 (cid:3)0
b
where Y ,Z ,Z ∈ C3n×n are arbitra(cid:2)ry. This co(cid:3)mpletes the proof.
1 1 2 b
Example 2.6 In Example 2.4 we achieve the linear two-parameter polynomial
L(λ,µ) ∈ L(Q(λ,µ)) by choosing
−A −A 0
20 01
Y := −A +A ,Z := A ,Z := A .
1 00 11 1 10 2 01
A −I +2A A
02 10 01
6
The standard linearization L(λ,µ) ∈ L(Q(λ,µ)) is achieved by choosing
0 A A
10 01
Y := 0 ,Z := 0 ,Z := −I .
1 1 2
0 −I 0
Corollary 2.7 Dimension of L(Q(λ,µ)) = 9n2 +3.
Remark 2.8 For quadratic one-parameter matrix polynomial
P(λ) = λ2A +λA +A ,A ∈ Cn×n,i = 0,1,2, (12)
20 10 00 i0
a vector space L (P) of matrix pencils of the form L(λ) = X +λY ∈ C2n×2n is
1
obtained in [21]. Setting µ = 0 in Q(λ,µ) we have Q(λ,0) = P(λ). Then from
the constructions of linear two-parameter polynomials given in Theorem 2.5 it
is easy to check that L(Q(λ,µ)) = L (P). In fact, if µ = 0 then L(Q(λ,µ))
1
contains matrix pencils L(λ) = λA +A ∈ C2n×2n where
1 3
A = v ⊗A −Z +v ⊗A ,A = Z v ⊗A ,
1 20 1 b b 10 3 1 00
v ∈ C2 and Z ∈(cid:2)C2n×n is arbitrary. Thu(cid:3)s we ob(cid:2)tain the sam(cid:3)e vector space
b1 b
of matrix pencils obtained in [21] for a given quadratic one-parameter matrix
polynomial Q(λ,0) = λ2A +λA +A = P(λ).
20 10 00
2.1 Construction of linearizations
It is not very clear that whether all linear two-parameter matrix polynomials
in the space L(Q(λ,µ)) are linearizations of Q(λ,µ). For example, consider any
L(λ,µ) ∈ L(Q(λ,µ)) corresponding to ansatz vector v = 0. Thus given a
quadratic two-parameter matrix polynomial Q(λ,µ) we need to identify which
L(λ,µ) in L(Q(λ,µ)) are linearizations.
We begin with a result concerning the special case of the ansatz vector
v = αe where e = 1 0 0 T and 0 6= α ∈ C.
1 1
Theorem 2.9 Let Q(cid:2)(λ,µ) =(cid:3) λ2A + λµA + µ2A +λA + µA + A be
20 11 02 10 01 00
a quadratic two-parameter matrix polynomial with real or complex coefficient
matrices of order n×n. Suppose L(λ,µ) = λA +µA +A ∈ L(Q(λ,µ)) with
1 2 3
respect to the ansatz vector v = αe ∈ C3, where
1
b b b
A = αe ⊗A −Y +αe ⊗A −Z +αe ⊗A
1 1 20 1 1 11 1 1 10
A2 = (cid:2)Y1 αe1 ⊗A02 −Z2 +αe1 ⊗A01 (cid:3)
b
A3 = (cid:2)Z1 Z2 αe1 ⊗A00 , (cid:3)
b
Y (cid:2) Z Z (cid:3)
11b 11 12 Z Z
Y = 0 ,Z = Z ,Z = Z ∈ C3n×n, det 21 22 6= 0. Then
1 1 21 2 22 Z Z
0 Z Z (cid:20) 31 32(cid:21)
31 32
L(λ,µ)is alinearizationof Q(λ,µ).
7
Proof: ByTheorem2.5, anylineartwo-parametermatrixpolynomialL(λ,µ) =
λA +µA +A ∈ L(Q(λ,µ)) corresponding to the ansatz vector v = αe is of
1 2 3 1
the form
b b bαA −Y +αA −Z +αA Y αA −Z +αA
20 11 11 11 10 11 02 12 01
L(λ,µ) = λ 0 −Y −Z +µ Y 0 −Z
21 21 21 22
0 −Y −Z Y 0 −Z
31 31 31 32
Z Z αA
11 12 00
+ Z Z 0 .
21 22
Z Z 0
31 32
Thus we have
W (λ,µ) W (λ,µ) W (λ,µ)
1 2 3
L(λ,µ) = µY +Z −λY +Z −λZ −µZ
21 21 21 22 21 22
µY +Z −λY +Z −λZ −µZ
31 31 31 32 31 32
where W (λ,µ) = αλA +µY +Z ,W (λ,µ) = αµA +λαA −λY +Z
1 20 11 11 2 02 11 11 12
and W (λ,µ) = αλA −λZ +αµA −µZ +αA .
3 10 11 01 12 00
Define
λI I 0
α
E(λ,µ) = µI 0 I .
α
1I 0 0
α
Consequently, we have
Q(λ,µ) W (λ,µ) W (λ,µ)
1 2
L(λ,µ)E(λ,µ) = 0 µY +Z −λY +Z .
21 21 21 22
0 µY +Z −λY +Z
31 31 31 32
Q(λ,µ) W(λ,µ)
Setting Y = 0 = Y we have L(λ,µ)E(λ,µ) = where
21 31 0 Z
(cid:20) (cid:21)
Z Z
W(λ,µ) = W (λ,µ) W (λ,µ) ∈ Cn×2n,Z = 21 22 ∈ C2n×2n.
1 2 Z Z
31 32
(cid:20) (cid:21)
Since Z(cid:2)is nonsingular, we d(cid:3)efine
I −W(λ,µ)Z−1
F(λ,µ) = .
0 Z−1
(cid:20) (cid:21)
Then we have
Q(λ,µ) 0
F(λ,µ)L(λ,µ)E(λ,µ) = .
0 I
2n
(cid:20) (cid:21)
Note that both E(λ,µ) and F(λ,µ) are unimodular polynomials. Hence we
have detL(λ,µ) = γdetQ(λ,µ) for some nonzero γ ∈ C. Thus L(λ,µ) is a
linearization. This completes the proof.
LetQ(λ,µ)quadratictwo-parametermatrixpolynomialandL(λ,µ) ∈ L(Q(λ,µ))
corresponding to anansatz vector 0 6= v ∈ C3. Then thefollowing isa procedure
for determining a set of linearizations of Q(λ,µ).
Procedure to determine linearizations in L(Q(λ,µ)):
8
1. Suppose Q(λ,µ) is a quadratic two-parameter matrix polynomial and
L(λ,µ) = λA + µA + A ∈ L(Q(λ,µ)) corresponding to ansatz vec-
1 2 3
tor v ∈ C3 i.e. L(λ,µ)(Λ⊗I ) = v⊗Q(λ,µ).
n
b b b
m m m
11 12 13
2. Select any nonsingular matrix M = m m m such that Mv =
21 22 23
m m m
31 32 33
αe ∈ C3,α 6= 0. A list of nonsingular matrices Mdepending on the
1
entries of v is given in the Appendix.
3. Apply the corresponding block-transformation M ⊗ I to L(λ,µ). Then
n
we have L(λ,µ) = (M ⊗I )L(λ,µ) = λA +µA +A such that
n 1 2 3
Ae = αe ⊗A −Y +αe ⊗fA f−Z f+αe ⊗A
1 1 20 1 1 11 1 1 10
h i
Af2 = Y1 αe1 ⊗A02e −Z2 +αe1 ⊗A01f
h i
Af3 = Ze1 Z2 αe1 ⊗A00f
h i
where f f f
Y m Y
11 11 11
Y = (M ⊗I )Y = (M ⊗I ) 0 = m Y
1 n 1 n 21 11
0 m Y
31 11
e Z m Z +m Z +m Z
11 11 11 12 21 13 31
Z = (M ⊗I )Z = (M ⊗I ) Z = m Z +m Z +m Z
1 n 1 n 21 21 11 22 21 23 31
Z m Z +m Z +m Z
31 31 11 32 21 33 31
f Z m Z +m Z +m Z
12 11 12 12 22 13 32
Z = (M ⊗I )Z = (M ⊗I ) Z = m Z +m Z +m Z
2 n 2 n 22 21 12 22 22 23 32
Z m Z +m Z +m Z
32 31 12 32 22 33 32
f
are arbitrary.
4. For L(λ,µ) to be linearization we need to choose Y ,Z ,Z as follows.
1 1 2
If m = m = 0 then choose Y arbitrary; otherwise choose Y = 0.
21 31 11 11
e Z Z
11 12
Further we need to choose Z = Z ,Z = Z for which
1 21 2 22
Z Z
31 32
m Z +m Z +m Z m Z +m Z +m Z
det 21 11 22 21 23 31 21 12 22 22 23 32 6= 0. (13)
m Z +m Z +m Z m Z +m Z +m Z
31 11 32 21 33 31 31 12 32 22 33 32
(cid:20) (cid:21)
From the construction of M given in the Appendix it is easy to check
that we can always choose suitable Z ,Z for which the condition (13) is
1 2
satisfied.
9
3 Linearization of two-parameter quadratic eigen-
value problem
The quadratic two-parameter eigenvalue problem is concerned with finding a
pair (λ,µ) ∈ C×C and nonzero vectors x ∈ Cni for which
i
Q (λ,µ)x = 0,i = 1,2, (14)
i i
where
Q (λ,µ) = A +λB +µC +λ2D +λµE +µ2F , (15)
i i i i i i i
A ,B ,...,F ∈ Cni×ni. The pair (λ,µ) is called an eigenvalue of (14) and x ⊗
i i i 1
x is called the corresponding eigenvector. The spectrum of a quadratic two-
2
parameter eigenvalue problem is the set
σ := {(λ,µ) ∈ C×C : detQ (λ,µ) = 0,i = 1,2}. (16)
Q i
In the generic case, we observe that (14) has 4n n eigenvalues by using the
1 2
following theorem.
Theorem 3.1 (Bezout’s Theorem, [3]) Let f(x,y) = g(x,y) = 0 be a system of
two polynomial equations in two unknowns. If it has only finitely many common
complex zeros (x,y) ∈ C×C, then the numberof those zeros is at most degree(f)·
degree(g).
The usual approach to solving (14) is to linearize it as a two-parameter
eigenvalue problem given by
L (λ,µ)w = (A(1))+λB(1) +µC(1))w = 0
1 1 1 (17)
L (λ,µ)w = (A(2))+λB(2) +µC(2))w = 0
2 2 2
(cid:26)
where A(i),B(i),C(i) ∈ Cmi×mi,m ≥ 2n ,i = 1,2, and w = Λ ⊗ x . A pair
i i i i
(λ,µ) is called an eigenvalue of (17) if L (λ,µ)w = 0 for a nonzero vector w
i i i
for i = 1,2, and w ⊗w is the corresponding eigenvector. Thus the spectrum
1 2
of the linearized two-parameter eigenvalue problem is given by
σ := {(λ,µ) ∈ C×C : detL (λ,µ) = 0,i = 1,2}. (18)
L i
Therefore, in the generic case, the problem (17) has m m ≥ 4n n eigenvalues.
1 2 1 2
A standard approach to solve a two-parameter eigenvalue problem (17) is
by converting it into a coupled generalized eigenvalue problem given by
△ z = λ△ z and △ z = µ△ z (19)
1 0 2 0
where z = w ⊗w and
1 2
△ = B(1) ⊗C(2) −C(1) ⊗B(2)
0
△ = C(1) ⊗A(2) −A(1) ⊗C(2)
1
△ = A(1) ⊗B(2) −B(1) ⊗A(2).
2
10
