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Linear Algebra and its Applications
journal homepage:w w w . e l s e v i e r . c o m / l o c a t e / l a aComplex symmetric stabilizing solution of the matrix
equation X
+
A
X
−
1
A
=
Q
Chun-Hua Guo
a,1, Yueh-Cheng Kuo
b,2, Wen-Wei Lin
c,d,∗
,3aDepartment of Mathematics and Statistics, University of Regina, Regina, Saskatchewan, Canada S4S 0A2 bDepartment of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan cDepartment of Mathematics, National Taiwan University, Taipei 106, Taiwan
dCenter of Mathematical Modelling and Scientific Computing, National Chiao Tung University, Hsinchu 300, Taiwan
A R T I C L E I N F O A B S T R A C T Article history:
Received 5 May 2010 Accepted 16 March 2011 Available online 20 April 2011 Submitted by V. Mehrmann
AMS classification:
15A24 65F30
Keywords:
Nonlinear matrix equation Complex symmetric solution Stabilizing solution Doubling algorithm
We study the matrix equation X+AX−1A=Q , where A is a com-plex square matrix and Q is comcom-plex 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 ap-plications, the existence of a unique complex symmetric stabilizing solution has been proved using advanced results on linear opera-tors. 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.
© 2011 Elsevier Inc. All rights reserved.
1. Introduction
The matrix equation X
+
A∗X−1A=
Q , where Q is Hermitian positive definite, arises in several applications. The corresponding real case is the matrix equation X+
AX−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.∗ Corresponding author at: Department of Mathematics, National Taiwan University, Taipei 106, Taiwan.
E-mail addresses:[email protected](C.-H. Guo),[email protected](Y.-C. Kuo),[email protected](W.-W. 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 and the National Center for Theoretical Sciences in Taiwan.
0024-3795/$ - see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.laa.2011.03.034
Recently, there arises the need to consider the matrix equation
X
+
AX−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= η
I for apos-itive 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 stabilizing solution has been proved using advanced results on linear operators (see [4, Chapter XXIV, Theorem 4.1,12]). 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 Eq. (1) we write
A
=
A1+
iA2,
Q=
Q1+
iQ2 (2)with A1, A2, Q1
=
Q1, Q2=
Q2∈ R
n×n. A solution X of (1) is said to be stabilizing ifρ(
X−1A) <
1,where
ρ(·)
denotes the spectral radius. The assumption we need to guarantee the existence of a stabilizing solution isQ2
+
eiθA2+
e−iθA2>
0,
forθ ∈ [
0,
2π].
(3)Here W
>
0 denotes the positive definiteness of a Hermitian matrix W . This assumption is satis-fied for the two applications we mentioned earlier. In particular, the assumption is trivially satissatis-fied for the nano application since A2=
0 and Q2= η
I withη >
0 there. Note that we do not needany further assumptions 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
+
A2Y−1A2=
Q2 has a realsym-metric 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 MJM=
LJL, where J=
⎡ ⎣ 0 I−
I 0 ⎤ ⎦. The matrix pair(
M,
L)
or the matrix pencil M− λ
L is called−
symplectic. It holds thatλ
is an eigenvalue of(
M,
L)
if and only if 1/λ
is an eigenvalue of(
M,
L)
, with the same multiplicity. Hereλ
can be 0 or∞
.Proof. We show that M
−
eiθL is nonsingular for allθ ∈ [
0,
2π]
. Suppose there are aθ
0∈ [
0,
2π]
and a nonzero vector x
=
x1,
x2 with x1,
x2∈ C
nsuch that(
M−
eiθ0L)
x=
0. This implies thatAx1
=
eiθ0x2,
Qx1−
x2=
eiθ0Ax1.
(5) By eliminating x2in (5) we have Hx1≡
eiθ0A−
Q+
e−iθ0A x 1=
0.
(6)Write H
=
H1+
iH2, where H1=
eiθ0A1−
Q1+
e−iθ0A1and H2=
eiθ0A2−
Q2+
e−iθ0A2. It is easilyseen that H1and H2are Hermitian. From assumption (3) it holds that H2is negative definite. By the
classical Bendixson theorem (see [14] for example) H1
+
iH2is invertible. From (6) and (5) it followsthat x1
=
0 and x2=
0. Thus, M−
eiθL is nonsingular for allθ ∈ [
0,
2π].
From Lemma1we see that there is a matrix ⎡ ⎣U
V ⎤
⎦
∈ C
2n×nof full rank spanning the stable invariantsubspace of M
− λ
L corresponding to the stable eigenvalue matrix S∈ C
n×n, i.e., ⎡ ⎣A 0 Q−
I ⎤ ⎦ ⎡ ⎣U V ⎤ ⎦=
⎡ ⎣ 0 I A 0 ⎤ ⎦ ⎡ ⎣U V ⎤ ⎦ S,
(7)where
ρ(
S) <
1. From (7) we getAU
=
VS,
(8)QU
−
V=
AUS.
(9)Multiplying (9) by U∗from the left we get U∗QU
−
U∗V=
U∗AUS=
U∗
A∗
+
2iA2US
.
(10)Substituting (8) into (10), we have
U∗QU
−
U∗V=
S∗V∗US+
2iU∗A2US.
(11) Taking conjugate transposes in (11) and subtracting the result from (11) we obtain2iU∗Q2U
+ (
V∗U−
U∗V) =
S∗(
V∗U−
U∗V)
S+
2i U∗A2US+
S∗U∗A2U.
(12) Let K=
i(
V∗U−
U∗V).
(13) Then K is Hermitian. From (12) it follows that K satisfies the equationK
−
S∗KS=
2U∗Q2U−
U∗A2US−
S∗U∗A2U
.
(14)Lemma 2. The matrix K in (13) is positive definite.
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)=
2 U∗Q2U+
S∗U∗Q2US+ · · · + (
S∗)
−1U∗Q2US−1−
U∗A2US−
S∗U∗A2US2− · · · − (
S∗)
−1U∗A2US−
S∗U∗A2U− (
S∗)
2U∗A2US− · · · − (
S∗)
U∗A2US−1.
(16)Since
ρ(
S) <
1, S→
0 as→ ∞
. Hence from (16) we have K=
2Q2−
A∗2S−
S∗A2,
(17) where Q2=
∞ =0(
S∗)
U∗Q2US,
A2=
∞ =0(
S∗)
U∗A2US.
(18)Note that Q2
+
eiθA2+
e−iθA2>
0 for allθ ∈ [
0,
2π]
is equivalent to thatA2
=
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ Q2−
A2−
A2 Q2−
A2... ... ...
... ...
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (19)is positive definite. From (17) and (18) it is easy to check that
K
=
2U∗,
S∗U∗, · · ·
A2 ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ U US...
⎤ ⎥ ⎥ ⎥ ⎥ ⎦.
(20)We need to show that z∗Kz
>
0 for all z=
0. SinceA2 is positive definite, it is enough to showWz
=
0 for all z=
0, where W is the rightmost block matrix in (20). Suppose Wz=
0. Then Uz=
0 and USz=
0. It follows from (9) that Vz=
QUz−
AUSz=
0. Thus⎡ ⎣U
V ⎤
⎦ z
=
0 and then z=
0 since ⎡⎣U V ⎤
⎦ is of full rank.
The next result follows readily.
Theorem 3. The matrix U in (7) is invertible.
Proof. Suppose Ux
=
0 with x∈ C
n. From (13) we have x∗Kx=
x∗iV∗U−
U∗V x=
0.
So x
=
0 since K is positive definite by Lemma2. Thus U is invertible. Since U is invertible, we can define X=
VU−1.Theorem 4. Let X
=
VU−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 Ufrom the left we get
Subtracting the transpose of (21) from (21) and using (8) we have UV
−
VU=
SUAU−
UAUS=
SUVS−
SVUS=
SUV−
VU S.
(22) Sinceρ(
S) <
1, UV=
VU. Then X=
VU−1=
U−(
UV)
U−1is a complex symmetric matrix.(b) From (8) and (9), and noting that UV
=
VU, we haveλ
2A− λ
Q+
A=λ
2U−SUVU−1− λ
VU−1+
U−SUVSU−1+
VSU−1=
I− λ
USU−1 VU−1−λ
I+
USU−1.
(23)Since det
(λ
2A− λ
Q+
A) =
det(
M− λ
L) =
0 for every unimodularλ
(by Lemma1), we know that X=
VU−1is nonsingular.(c) From (8) and (9) we have
A
=
X(
USU−1),
Q−
X=
A(
USU−1).
(24) Eliminating USU−1in (24) gives X+
AX−1A=
Q and we also haveρ(
X−1A) = ρ(
USU−1) = ρ(
S) <
1.(d) From (13) it follows that
U−∗KU−1
=
iX∗−
X=
2Im(
X).
(25) So X2≡
Im(
X)
is positive definite by Lemma2.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 Eq. (1) also have a positive definite imaginary part. In fact, for a real matrix A2and a real symmetric positive definite matrix Q2satisfying the assumption (3), the
equation Y
+
AT2Y−1A2=
Q2may have many positive definite solutions Y (see [3]). So for each suchY , X
=
iY is a solution of X+ (
iA2)
X−1(
iA2) =
iQ2with 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 A2and a real symmetric positive definite matrix Q2
satisfy-ing the assumption (3), the equation Y
+
AT2Y−1A2=
Q2has a positive definite stabilizing solution Y .In fact, we have already proved that the equation X
+ (
iA2)
TX−1(
iA2) =
iQ2has a complex symmetricstabilizing solution X with a positive definite imaginary part. We only need to show that the real part of X must be zero. Since A
=
iA2and Q=
iQ2now, we have from (10) and (8) thatU∗QU
−
U∗V= −
U∗A∗US= −
S∗V∗US.
(26) Taking conjugate transpose on (26) gives−
U∗QU−
V∗U= −
S∗U∗VS.
(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 sinceρ(
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) complex 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 Eq. (1) with the matrices A and Q having special block structures. However, 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 Eq. (1) with assumption (3).
Acknowledgements
The authors thank Beatrice Meini and two referees for their helpful comments. In particular, com-ments by one referee significantly simplified the argucom-ments leading to the conclusion in Theorem3.
References
[1] C.-Y. Chiang, E.K.-W. Chu, C.-H. Guo, T.-M. Huang, W.-W. Lin, S.-F. Xu, Convergence analysis of the doubling algorithm for several nonlinear matrix equations in the critical case, SIAM J. Matrix Anal. Appl. 31 (2009) 227–247.
[2] S. Datta, Nanoscale device modeling: the Green’s function method, Superlattices and Microstructures 28 (2000) 253–278. [3] J.C. Engwerda, A.C.M. Ran, A.L. Rijkeboer, Necessary and sufficient conditions for the existence of a positive definite solution
of the matrix equation X+A∗X−1A=Q , Linear Algebra Appl. 186 (1993) 255–275.
[4] I. Gohberg, S. Goldberg, M.A. Kaashoek, Classes of Linear Operators, vol. II, Operator Theory: Advances and Applications, vol. 63, Birkhäuser, 1993.
[5] C.-H. Guo, P. Lancaster, Iterative solution of two matrix equations, Math. Comp. 68 (1999) 1589–1603.
[6] C.-H. Guo, W.-W. Lin, The matrix equation X+AX−1A=Q and its application in nano research, SIAM J. Sci. Comput. 32
(2010) 3020–3038.
[7] C.-H. Guo, W.-W. Lin, Solving a structured quadratic eigenvalue problem by a structure-preserving doubling algorithm, SIAM J. Matrix Anal. Appl. 31 (2010) 2784–2801.
[8] D.L. John, D.L. Pulfrey, Green’s function calculations for semi-infinite carbon nanotubes, Phys. Status Solidi B Basic Solid State Phys. 243 (2006) 442–448.
[9] A. Kletsov, Y. Dahnovsky, J.V. Ortiz, Surface Green’s function calculations: a nonrecursive scheme with an infinite number of principal layers, J. Chem. Phys. 126 (2007) 5 (Article No. 134105).
[10] W.-W. Lin, S.-F. Xu, Convergence analysis of structure-preserving doubling algorithms for Riccati-type matrix equations, SIAM J. Matrix Anal. Appl. 28 (2006) 26–39.
[11] D.S. Mackey, N. Mackey, C. Mehl, V. Mehrmann, Structured polynomial eigenvalue problems: good vibrations from good linearizations, SIAM J. Matrix Anal. Appl. 28 (2006) 1029–1051.
[12] C. van der Mee, G. Rodriguez, S. Seatzu, LDU factorization results for bi-infinite and semi-infinite scalar and block Toeplitz matrices, Calcolo 33 (1996) 307–335.
[13] B. Meini, Efficient computation of the extreme solutions of X+A∗X−1A=Q and X−A∗X−1A=Q , Math. Comp. 71 (2002) 1189–1204.
[14] H. Wielandt, On eigenvalues of sums of normal matrices, Pacific J. Math. 5 (1955) 633–638. [15] X. Zhan, J. Xie, On the matrix equation X+AX−1A=I, Linear Algebra Appl. 247 (1996) 337–345.
