Complex symmetric stabilizing solution of the
matrix equation X + A
>X
−1A = Q
Chun-Hua Guo
a,1, Yueh-Cheng Kuo
b,2, Wen-Wei Lin
c,∗
,3aDepartment of Mathematics and Statistics, University of Regina, Regina,
Saskatchewan S4S 0A2, Canada
bDepartment of Applied Mathematics, National University of Kaohsiung,
Kaohsiung 811, Taiwan
cDepartment of Mathematics, National Taiwan University, Taipei 106, Taiwan &
Center of Mathematical Modelling and Scientific Computing, National Chiao Tung University, Hsinchu 300, Taiwan
Abstract
We study the matrix equation X + A>X−1A = Q, where A is a complex square ma-trix and Q is complex symmetric. Special cases of this equation appear in Green’s function calculation in nano research and also in the vibration analysis of fast trains. In those applications, the existence of a unique complex symmetric stabilizing so-lution has been proved using advanced results on linear operators. The stabilizing solution is the solution of practical interest. In this paper we provide an elementary proof of the existence for the general matrix equation, under an assumption that is satisfied for the two special applications. Moreover, our new approach here re-veals that the unique complex symmetric stabilizing solution has a positive definite imaginary part. The unique stabilizing solution can be computed efficiently by the doubling algorithm.
AMS classification: 15A24, 65F30
Key words: nonlinear matrix equation, complex symmetric solution, stabilizing solution, doubling algorithm
∗ Corresponding author.
Email addresses: [email protected] (Chun-Hua Guo),
[email protected] (Yueh-Cheng Kuo), [email protected] (Wen-Wei Lin).
1 The work of this author was supported in part by a grant from the Natural
Sciences and Engineering Research Council of Canada.
2 The work of this author was partially supported by the National Science Council
in Taiwan.
3 The work of this author was partially supported by the National Science Council
1 Introduction
The matrix equation X + A∗X−1A = Q, where Q is Hermitian positive defi-nite, arises in several applications. The corresponding real case is the matrix equation X + A>X−1A = Q, where A is real and Q is real symmetric positive definite. In both cases, we may assume without loss of generality that Q = I, the identity matrix. These equations have been studied in [1,3,5,10,13,15], for example.
Recently, there arises the need to consider the matrix equation
X + A>X−1A = Q, (1)
where A is complex and Q is complex symmetric. First, it is explained in [6] that the computation of the surface Green’s function in nano research [2,8,9] can be reduced to the problem of solving the matrix equation (1), where Q = Q1+ iQ2 with Q1 real symmetric and Q2 = ηI for a positive scalar η, but
the matrix A is still a real matrix. And then it is shown in [7] that a quadratic eigenvalue problem arising from the vibration analysis of fast trains [11] can be solved efficiently and accurately by solving a matrix equation of the form (1), where A is complex and Q is complex symmetric.
In those two applications, the existence of a unique complex symmetric sta-bilizing solution has been proved using advanced results on linear operators (see [4, Chapter XXIV, Theorem 4.1] and [12]). The stabilizing solution is the solution of practical interest. In Section 2 we provide an elementary proof of the existence for the general matrix equation (1), under an assumption that is satisfied for the two special applications. Moreover, our new approach reveals that the unique complex symmetric stabilizing solution has a positive definite imaginary part. In Section 3 we make some concluding remarks. In particular, we mention that the unique stabilizing solution can be computed efficiently by the doubling algorithm, as for the special case studied in [7].
2 Existence of complex symmetric stabilizing solution
For equation (1) we write
A = A1+ iA2, Q = Q1+ iQ2 (2)
with A1, A2, Q1 = Q1>, Q2 = Q>2 ∈ Rn×n. A solution X of (1) is said to
assumption we need to guarantee the existence of a stabilizing solution is Q2+ eiθA>2 + e
−iθ
A2 > 0, for θ ∈ [0, 2π]. (3)
Here W > 0 denotes the positive definiteness of a Hermitian matrix W . This assumption is satisfied for the two applications we mentioned earlier. In partic-ular, the assumption is trivially satisfied for the nano application since A2 = 0
and Q2 = ηI with η > 0 there. Note that we do not need any further
assump-tions on the matrices A1 and Q1. Also, if (3) has been verified for the matrices
A2 and Q2, then it also holds when any positive semi-definite matrix is added
to Q2. From [3] we also know that (3) holds if and only if the matrix equation
Y + A>2Y−1A2 = Q2 has a real symmetric positive definite stabilizing solution
Y . So one way to verify the assumption (3) is to use the doubling algorithm in [10] or the equivalent cyclic reduction algorithm in [13] to find the stabilizing solution Y .
We now assume (3) and let
M = A 0 Q −I , L = 0 I A> 0 . (4)
It is easily seen that the matrix pair (M, L) satisfies the relation M J M> = LJ L>, where J = 0 I −I 0
. The matrix pair (M, L) or the matrix pencil M − λL is called >−symplectic. It holds that λ is an eigenvalue of (M, L) if and only if 1/λ is an eigenvalue of (M, L), with the same multiplicity. Here λ can be 0 or ∞.
Lemma 1 The >−symplectic pencil M − λL has no eigenvalues on the unit circle.
PROOF. We show that M − eiθL is nonsingular for all θ ∈ [0, 2π]. Suppose there are a θ0 ∈ [0, 2π] and a nonzero vector x = (x>1, x
> 2)
> with x
1, x2 ∈ Cn
such that (M − eiθ0L)x = 0. This implies that
Ax1 = eiθ0x2, Qx1− x2 = eiθ0A>x1. (5) By eliminating x2 in (5) we have Hx1 ≡ eiθ0A>− Q + e−iθ0Ax 1 = 0. (6)
Write H = H1+ iH2, where H1 = eiθ0A>1 − Q1+ e−iθ0A1 and H2 = eiθ0A>2 −
Q2+ e−iθ0A2. It is easily seen that H1 and H2are Hermitian. From assumption
(see [14] for example) H1+ iH2 is invertible. From (6) and (5) it follows that
x1 = 0 and x2 = 0. Thus, M − eiθL is nonsingular for all θ ∈ [0, 2π]. 2
From Lemma 1 we see that there is a matrix U V ∈ C 2n×n of full rank
spanning the stable invariant subspace of M − λL corresponding to the stable eigenvalue matrix S ∈ Cn×n, i.e.,
A 0 Q −I U V = 0 I A> 0 U V S, (7)
where ρ(S) < 1. From (7) we get
AU = V S, (8)
QU − V = A>U S. (9)
Multiplying (9) by U∗ from the left we get
U∗QU − U∗V = U∗A>U S = U∗(A∗+ 2iA>2)U S. (10) Substituting (8) into (10), we have
U∗QU − U∗V = S∗V∗U S + 2iU∗A>2U S. (11) Taking conjugate transpose in (11) and subtracting the result from (11) we obtain 2iU∗Q2U + (V∗U − U∗V ) = S∗(V∗U − U∗V )S + 2i(U∗A>2U S + S ∗ U∗A2U ). (12) Let K = i(V∗U − U∗V ). (13)
Then K is Hermitian. From (12) it follows that K satisfies the equation K − S∗KS = 2(U∗Q2U − U∗A>2U S − S
∗
U∗A2U ). (14)
PROOF. From (14), for any positive integer ` we have K − (S∗)`KS` =(K − S∗KS) + S∗(K − S∗KS)S + · · · + (S∗)`−1(K − S∗KS)S`−1 (15) =2hU∗Q2U + S∗U∗Q2U S + · · · + (S∗)`−1U∗Q2U S`−1 − U∗A>2U S − S∗U∗A>2U S2− · · · − (S∗)`−1U∗A>2U S` −S∗U∗A2U − (S∗)2U∗A2U S − · · · − (S∗)`U∗A2U S`−1 i . (16) Since ρ(S) < 1, S` → 0 as ` → ∞. Hence from (16) we have
K = 2(Qe2 −Ae∗ 2S − S ∗ e A2), (17) where e Q2 = ∞ X `=0 (S∗)`U∗Q2U S`, Ae2 = ∞ X `=0 (S∗)`U∗A2U S`. (18)
Note that Q2+ eiθA>2 + e −iθA
2 > 0 for all θ ∈ [0, 2π] is equivalent to that
A2 = Q2 −A>2 −A2 Q2 −A>2 . .. . .. ... . .. ... (19)
is positive definite. From (17) and (18) it is easy to check that
K = 2 [U∗, S∗U∗, · · · ] A2 U U S .. . . (20)
We need to show that z∗Kz > 0 for all z 6= 0. Since A2 is positive definite,
it is enough to show W z 6= 0 for all z 6= 0, where W is the rightmost block matrix in (20). Suppose W z = 0. Then U z = 0 and U Sz = 0. It follows from (9) that V z = QU z − A>U Sz = 0. Thus U V
z = 0 and then z = 0 since U V is of full rank. 2 The next result follows readily.
Theorem 3 The matrix U in (7) is invertible.
PROOF. Suppose U x = 0 with x ∈ Cn. From (13) we have
x∗Kx = x∗[i (V∗U − U∗V )] x = 0.
So x = 0 since K is positive definite by Lemma 2. Thus U is invertible. 2 Since U is invertible, we can define X = V U−1.
Theorem 4 Let X = V U−1. Then (a) X is complex symmetric; (b) X is invertible;
(c) X is a stabilizing solution of (1); (d) X2 ≡ Im(X) is positive definite.
PROOF. (a) Multiplying (9) by U> from the left we get
U>QU − U>V = U>A>U S. (21) Subtracting the transpose of (21) from (21) and using (8) we have
U>V − V>U = S>U>AU − U>A>U S
= S>U>V S − S>V>U S = S>U>V − V>US. (22) Since ρ(S) < 1, U>V = V>U . Then X = V U−1 = U−>(U>V )U−1 is a complex symmetric matrix.
(b) From (8) and (9), and noting that U>V = V>U , we have λ2A>− λQ + A
=λ2U−>S>U>V U−1− λV U−1+ U−>S>U>V SU−1+ V SU−1 =I − λU SU−1>V U−1−λI + U SU−1. (23) Since det(λ2A>− λQ + A) = det(M − λL) 6= 0 for every unimodular λ (by
Lemma 1), we know that X = V U−1 is nonsingular. (c) From (8) and (9) we have
A = X(U SU−1), Q − X = A>(U SU−1). (24) Eliminating U SU−1in (24) gives X+A>X−1A = Q and we also have ρ(X−1A) = ρ(U SU−1) = ρ(S) < 1.
(d) From (13) it follows that
U−∗KU−1 = i (X∗− X) = 2Im(X). (25) So X2 ≡ Im(X) is positive definite by Lemma 2. 2
We have shown that the (unique) stabilizing solution of (1) must be complex symmetric, and that it has a positive definite imaginary part. When A is not a real matrix, it is quite possible that some other complex symmetric solutions of the equation (1) also have a positive definite imaginary part. In fact, for a real matrix A2 and a real symmetric positive definite matrix Q2 satisfying the
assumption (3), the equation Y + AT
2Y−1A2 = Q2 may have many positive
definite solutions Y (see [3]). So for each such Y , X = iY is a solution of X + (iA2)X−1(iA2) = iQ2 with a positive definite imaginary part.
We can also provide an elementary proof for the following statement proved in [3] using advanced results in operator theory: for a real matrix A2 and a
real symmetric positive definite matrix Q2 satisfying the assumption (3), the
equation Y + AT 2Y
−1A
2 = Q2 has a positive definite stabilizing solution Y . In
fact, we have already proved that the equation X + (iA2)TX−1(iA2) = iQ2 has
a complex symmetric stabilizing solution X with a positive definite imaginary part. We only need to show that the real part of X must be zero. Since A = iA2
and Q = iQ2 now, we have from (10) and (8) that
U∗QU − U∗V = −U∗A∗U S = −S∗V∗U S. (26) Taking conjugate transpose on (26) gives
−U∗QU − V∗U = −S∗U∗V S. (27) It follows from (26) and (27) that
(U∗V + V∗U ) − S∗(U∗V + V∗U )S = 0. (28) So U∗V + V∗U = 0 since ρ(S) < 1. Now 2Re(X) = X + X∗ = U−∗(U∗V + V∗U )U−1 = 0.
3 Conclusions
We have provided an elementary proof of the existence of a (unique) com-plex symmetric stabilizing solution X for the nonlinear matrix equation (1) with assumption (3). Our new approach here has revealed that the imaginary part of X is positive definite. We also mention that the solution X can be
found efficiently by a doubling algorithm, as presented in [7, Algorithm 4.1]. A convergence result for the algorithm is given in [7, Theorem 4.1] for the equation (1) with the matrices A and Q having special block structures. How-ever, those special structures were not used in the proof of convergence in [7]. So the statements in that theorem are also valid for our general equation (1) with assumption (3).
Acknowledgement
The authors thank Beatrice Meini and two referees for their helpful comments. In particular, comments by one referee significantly simplified the arguments leading to the conclusion in Theorem 3.
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