A REAL-VALUED SPECTRAL DECOMPOSITION ... - Semantic Scholar
SIAM J. MATRIX ANAL. APPL. Vol. 32, No. 2, pp. 584–604
© 2011 Society for Industrial and Applied Mathematics
A REAL-VALUED SPECTRAL DECOMPOSITION OF THE UNDAMPED GYROSCOPIC SYSTEM WITH APPLICATIONS* ZHIGANG JIA†
AND
MUSHENG WEI‡
Abstract. Given an undamped gyroscopic system GðλÞ ¼ M λ2 þ Cλ þ K with M , K symmetric and C skew-symmetric, this paper presents a real-valued spectral decomposition of GðλÞ by a real standard pair ðX; T Þ and a skew-symmetric parameter matrix S. When T is assumed to be a block diagonal matrix, the parameter matrix S has a special structure. This spectral decomposition is applied to solve the quadratic inverse eigenvalue problem and the no spill-over quadratic eigenvalue updating problem. Key words. gyroscopic system, spectral decomposition, real standard pair, skew-Hankel matrix, quadratic inverse eigenvalue problem, no spill-over eigenvalue updating problem AMS subject classifications. 15A24, 15A29, 65F18 DOI. 10.1137/100792020
1. Introduction. Gyroscopic systems in design and analysis of vibrating structures, such as bridges, highways, buildings, airplanes, etc., can be defined by a homogenous distributed parameter system (see [2] and [5], for example). Very often a distributed parameter system is first discretized to a matrix second-order model using techniques of finite element or finite difference, and then the problem is solved for this discretized reduced-order model. Associated with the matrix second-order model is the eigenvalue problem of the quadratic pencil, ð1:1Þ
QðλÞ ¼ M λ2 þ ðD þ GÞλ þ K ;
where M ¼ M T is the mass or inertia matrix, K ¼ K T the stiffness matrix, D ¼ DT the damping matrix, and G ¼ −G T the gyroscopic matrix. (Here, AT denotes the transposed matrix of A.) In many applications one often encounters two important special cases of (1.1): the damped nongyroscopic system when G ¼ 0, and the undamped gyroscopic system when D ¼ 0. The damped nongyroscopic system has been widely studied in two aspects: the quadratic eigenvalue problem (QEP), and the quadratic inverse eigenvalue problem (QIEP). The QEP analyzes and computes the spectral information; hence it reveals the dynamical behavior of the system from a priori known physical parameters such as mass, elasticity, inductance, and capacitance. The QIEP determines or estimates the parameters of the system from observed or expected eigeninformation. For the damped nongyroscopic system, Lancaster [24] and Gohberg, Lancaster, and Rodman [20], [21] studied some theoretical results of the QEP; Tisseur and Meerbergen [32] provided a good survey of applications, mathematical properties, and variety of numerical algorithms of the QEP; Bai and Su [1] proposed a second-order Arnoldi *Received by the editors April 12, 2010; accepted for publication (in revised form) by P. Benner April 4, 2011; published electronically June 29, 2011. This work was supported by the NSFC under grant 10771073 and the Shanghai Leading Academic Discipline Project under grant S30405. http://www.siam.org/journals/simax/32-2/79202.html † Department of Mathematics, Xuzhou Normal University, Jiangsu 221116, China ([email protected]). ‡ Corresponding author. College of Mathematics and Science, Shanghai Normal University, Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China ([email protected]). 584
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method for the large-scale QEP; Chu and Golub [11] provided an excellent survey of standard QIEP; Lancaster and Prells [27] discussed the QIEP with all eigenvalues being simple and nonreal from three points of view: spectral theory, structure-preserving similarity transforms, and factorization properties; Chu and Xu [14] characterized a realvalued spectral decomposition for real symmetric quadratic λ-matrices and described its applications to three challenging inverse problems: the QIEP (with the entire eigeninformation being given), the total decoupling problem, and the eigenvalue embedding problem. With only partially prescribed eigenpairs available, Chu, Kuo, and Lin [12] put forward a special solution for the symmetric QIEP. Shortly afterwards, for the QIEP with 1 < k ≤ n prescribed eigenpairs, Kuo, Lin, and Xu [23] presented a general solution and some particular solutions with additional eigenstructures. For the QIEP with n < k ≤ 2n prescribed eigenpairs, Cai et al. [6] derived a general solution in the parameterized forms in terms of two constrained parameter matrices and gave some necessary and sufficient conditions to guarantee that mass matrix is nonsingular and symmetric positive definite. Recently, Lin, Dong, and Chu [29] described an application of semidefinite programming techniques to the QIEP; Datta [15] briefly reviewed recent development of the QIEP with applications to active vibration control and finite element model updating. Study and applications of the undamped gyroscopic system are as important as the damped nongyroscopic system. The λ-matrix of the undamped gyroscopic system can be denoted by ð1:2Þ
GðλÞ ¼ M λ2 þ C λ þ K ;
where M , K , C are real square matrices with M T ¼ M and nonsingular, K T ¼ K , C T ¼ −C . It is well known that the eigenvalues of (1.2) have a Hamiltonian structure; ¯ −λ; ¯ −λÞ, possibly collapsing to real or imaginary pairs i.e., they occur in quadruples ðλ; λ; or a single zero eigenvalue. The QEP of the undamped gyroscopic system can be represented in two ways, i.e., by linearization or nonlinear matrix equations. A skew-Hamiltonian/Hamiltonian linearization of GðλÞ is M C 0 −K LðλÞ ≔ Lðλ; M ; C ; K Þ ¼ λ− : 0 M M 0 It is easy to see that GðλÞ and LðλÞ have precisely the same eigenvalues. For the eigenvalue problem of LðλÞ, Benner, Mehrmann, and Xu [4], Chu, Liu, and Mehrmann [8], and others proposed some structure-preserving algorithms by means of symplectic orthogonal transformations which perform more efficiently than the QZ and QR algorithms; see [32] for a detailed discussion. Guo [22] proposed an algorithm for finding all eigenvalues of the QEP of GðλÞ with M positive definite and K negative definite, by solving the maximal solution of a nonlinear matrix equation Z þ AT Z −1 A ¼ Q, where A ¼ M þ C þ K and Q ¼ 2ðM − K Þ. The algorithm is quadratic convergent when the QEP has no purely imaginary eigenvalues. This algorithm preserves the Hamiltonian structure of the QEP. Qian and Lin [30] considered a more general case when the QEP of GðλÞ has eigenvalues on the imaginary axis. They first computed the purely imaginary eigenvalues by the Newton’s method [31], then shifted the purely imaginary eigenvalues with no spill-over to get a new gyroscopic system which has no eigenvalues on imaginary axis, and then finally applied the method proposed by Guo [22] to the new gyroscopic system to compute the remaining eigenvalues with quadratic convergence.
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One may solve the QIEP of the undamped gyroscopic system by a powerful GLR theory developed by Gohberg, Lancaster, and Rodman [21] for general matrix polynomials of arbitrary degrees. Lancaster [28] showed that a Jordan triple ðX; J; Y Þ for a quadratic matrix polynomial can generate a unique system ðM ; C ; K Þ. In general, given a standard pair ðX; T Þ or a Jordan pair ðX; J Þ, Y can be computed by solving a nonsingular matrix equation. To obtain a self-adjoint triple ðX; J; Y Þ, Y must have a required form. In order to avoid this difficulty Lancaster [28] proposed an efficient geometric/computational approach by solving a quadratic matrix equation. Notice that the Jordan triple ðX; J ; Y Þ constructed by the GLR theory for the spectral decomposition is complex-valued. Therefore, by applying the GLR theory to a real-valued undamped gyroscopic system, one may still obtain a complex-valued spectral decomposition. Thus it becomes very interesting and challenging to derive a real-valued spectral decomposition for a real-valued undamped gyroscopic system. Chu and Xu [14] developed an elegant procedure to obtain a real-valued spectral decomposition of the damped nongyroscopic system QðλÞ ¼ M λ2 þ C λ þ K , where M , C , and K are symmetric n × n matrices and M is nonsingular. For a real standard pair ðX; ℑÞ containing the spectral data, if there exists a real nonsingular matrix S such that S T ¼ S, ℑS ¼ ðℑSÞT , and XSX T ¼ 0, then ðM ; C ; K Þ can be characterized in terms of ðX; ℑÞ and S. With a natural assumption that ℑ is block diagonal, Chu and Xu [14] derived the structure of S. In general cases, S is a symmetric block diagonal matrix with its blocks or subblocks being partitioned into upper triangular Hankel blocks; in the special case when all eigenvalues are semisimple, there exists a real standard pair such that S has a simpler structure. Inspired by the analysis in [28], [14], in this paper we derive a real-valued spectral decomposition of the undamped gyroscopic system (1.2) in terms of a real standard pair ðX; T Þ and the parameter matrix S. Here S is block diagonal satisfying S T ¼ −S, T S ¼ ðT SÞT , and XSX T ¼ 0, its blocks are of four different (symmetric/skewsymmetric) upper triangular skew-Hankel forms defined in section 3. Therefore, the parameter matrix for the undamped gyroscopic system has a more complicated formula than that for the damped nongyroscopic system discussed in [14]. If all eigenvalues of the undamped gyroscopic system are semisimple, the diagonal blocks of S are of simpler structures. But the formula of S cannot be obtained using the method of Chu and Xu [14], since the blocks of S are not symmetric Hamiltonian. Instead, we use four different algorithms: the algorithm proposed in [4], the algorithm proposed in [8], the SVD, and the orthogonal decomposition. We also apply the real-valued spectral decomposition to solve the QIEP and the no spill-over quadratic eigenvalue updating problem of the undamped gyroscopic system (1.2). Especially, we prove that the no spill-over quadratic eigenvalue updating problem of (1.2) always has a solution, and complex eigenvalues with nonzero real part, purely imaginary eigenvalues, and nonzero real eigenvalues can be replaced by each other and preserve Hamiltonian property, which are not discussed in [14]. This paper is organized as follows. In section 2, we present a real-valued spectral decomposition of a real undamped gyroscopic system. In section 3, we characterize the structure of the parameter matrix S corresponding to some special real standard pairs. In section 4, we provide some applications and examples of the spectral decomposition to the QIEP and the no spill-over quadratic eigenvalue updating problem. In section 5, we make some concluding remarks. 2. The spectral decomposition. In this section, we establish the relationship between the coefficient matrices M , C , K in (1.2), a real standard pair ðX; T Þ, and
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UNDAMPED GYROSCOPIC SYSTEM
a skew-symmetric parameter matrix S. The analysis is motivated by the work of Chu and Xu [14] for the real damped nongyroscopic system. Recall from [21] that a matrix pair ðX; T Þ ∈ Rn×2n × R2n×2n is a real standard pair X is nonsingular, and the matrix of GðλÞ if and only if the matrix U ¼ U ðX; T Þ ≔ ½XT equation ð2:1Þ
M XT 2 þ C XT þ K X ¼ 0
holds. Equation (2.1) can also be written as C M X −K ð2:2Þ T¼ −M 0 XT 0
0 −M
X : XT
It is well known that the standard pair ðX; T Þ contains precisely the same spectral information of GðλÞ. Let Ψ be a given m × m real matrix. Define ð2:3Þ
S Ψ ¼ fS ∈ Rm×m jS ¼ −S T ; ΨS ¼ ðΨSÞT g:
THEOREM 2.1. Let ðX; T Þ ∈ Rn×2n × R2n×2n with U ðX; T Þ nonsingular. Then there exists an undamped gyroscopic system GðλÞ as in (1.2) with M nonsingular such that ðX; T Þ serves as a real standard pair if and only if there exists a nonsingular matrix S ∈ S T such that ð2:4Þ
XSX T ¼ 0:
In this case, the matrix triple ðM ; C ; K Þ is given by ð2:5Þ M ¼ ðXT SX T Þ−1 ;
C ¼ −M XT 2 SX T M ;
K ¼ −M XT 3 SX T M þ C M −1 C :
Proof. Necessity. Suppose that ðX; T Þ is a real standard pair of an undamped gyroscopic system GðλÞ. Because M is real symmetrical and nonsingular and C is real C M skew-symmetrical, the matrix ½−M 0 is real skew-symmetrical and nonsingular. Define a nonsingular skew-symmetric matrix −1 X T C M X S¼ ð2:6Þ : XT −M 0 XT Equation (2.2) implies that S −1 T
¼
UT
−K 0
0 U −M
is symmetric; therefore T S is symmetric and so S ∈ S T . Then from (2.6), X X T C I 0 0 X M T SX M ¼ ; ; S ¼ XT XT 0 I I XT −M 0 from which we obtain (2.4) and the formula M in (2.5). Postmultiplying (2.1) by SX T M and using the formula of M , we obtain the formula C in (2.5). Similarly, postmultiplying (2.1) by T SX T M and using the formulas of M and C , we obtain the formula of K in (2.5).
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Sufficiency. Suppose that there exists a nonsingular matrix S ∈ S T satisfying XSX T ¼ 0. Then the matrices defined in (2.5) possess properties that M , K are symmetric and C is skew-symmetric. We then observe that
X ½ T SX T M þ SX T C XT
SX T M ¼
I 0
0 ; I
and so M XT
2
X XT
−1
¼ M XT 2 ½ T SX T M þ SX T C
SX T M ¼ ½ −K
−C ;
which is equivalent to M XT 2 þ C XT þ K X ¼ 0. ▯ Remark 2.1. It can be proven that for S ∈ S T , T 2m S is skew-symmetric and T 2mþ1 S is symmetric for a nonnegative integer m. Suppose that the assertion is true for 0 ≤ m ≤ n. Then for m ¼ n þ 1, T 2nþ2 S ¼ T ðT 2nþ1 SÞT ¼ T ð−SÞðT 2nþ1 ÞT ¼ −ðT SÞT ðT 2nþ1 ÞT ¼ −ðT 2nþ2 SÞT ; T 2nþ3 S ¼ −T ðT 2nþ2 SÞT ¼ −T ð−SÞðT 2nþ2 ÞT ¼ ðT SÞT ðT 2nþ2 ÞT ¼ ðT 2nþ3 SÞT : 3. The structure of S. In this section, for given spectral data of GðλÞ we first construct a real standard pair ðX; T Þ, where T is defined as a block diagonal matrix, and we then characterize the special structure of the parameter matrix S corresponding to ðX; T Þ. Let J ðλj Þ ¼ λj I nj þ N j be the Jordan canonical form associated with the eigenvalue λj which may contain several Jordan blocks, and let X j be the n × nj matrix whose columns form the corresponding generalized eigenspace. Here, N j is an nj × nj nilpotent matrix with ones or zeros along its superdiagonal, depending on the partial multiplicities of λj . We now separate distinct eigenvalues λj and corresponding eigentriples of GðλÞ into the following four categories. Case 1. For j ¼ 1; : : : ; l1 ; λj ¼ αj þ iβj with αj , βj > 0. Then ½X jþ ; J ðλj Þ, ½X¯ jþ ; J ðλ¯ j Þ, ½X j− ; J ð−λ¯ j Þ, ½X¯ j− ; J ð−λj Þ are eigenpairs, where X jþ ¼ X jRþ þ iX jI þ , X j− ¼ X jR− þ iX jI − , X jR , X jI ∈ Rn×nj . Case 2. For j ¼ l1 þ 1; : : : ; l2 ; λj ¼ iβj with βj > 0. Then ½X j ; J ðλj Þ, ½X¯ j ; J ðλ¯ j Þ are eigenpairs, where X j ¼ X jR þ iX jI , X jR , X jI ∈ Rn×nj . Case 3. For j ¼ l2 þ 1; : : : ; k; λj ¼ αj with αj > 0. Then ½X jþ ; J ðλj Þ, ½X j− ; J ð−λj Þ are eigenpairs, where X jþ ¼ X jRþ , X j− ¼ X jR− ∈ Rn×nj . Case 4. λkþ1 ¼ 0. Then ½X kþ1 ; J ðλkþ1 Þ is an eigenpair, where X kþ1 ∈ Rn×2nkþ1 .1 Thus, 4ðn1 þ · · · þnl1 Þ þ 2ðnl1 þ1 þ · · · þnk þ nkþ1 Þ ¼ 2n, and we can construct a real standard pair ½X; T for GðλÞ as follows: X ≔ ½X 1Rþ ; X 1I þ ; X 1R− ; X 1I − ; : : : ; X l1 Rþ ; X l1 I þ ; X l1 R− ; X l1 I − ; X ðl1 þ1ÞR ; X ðl1 þ1ÞI ; : : : ; X l2 R ; X l2 I ; X ðl2 þ1ÞRþ ; X ðl2 þ1ÞR− ; : : : ; X kRþ ; X kR− ; X kþ1 ; ð3:1Þ
T ≔ diagðJ r ðλ1 Þ; : : : ; J r ðλkþ1 ÞÞ;
where J r ðλj Þ, j ¼ 1; : : : ; k þ 1, are, respectively, defined by 1
Notice that if zero is an eigenvalue of GðλÞ, its algebra multiplicity is even.
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UNDAMPED GYROSCOPIC SYSTEM
Jþ 0 r ðλj Þ J r ðλj Þ ≔ ; 0 J− r ðλj Þ ð3:2Þ
ð3:3Þ
J r ðλj Þ ≔
J r
αj I nj þ N j ¼ −βj I nj
Nj −βj I nj
βj I nj ; Nj
αj I nj þ N j J r ðλj Þ ≔ 0
ð3:4Þ ð3:5Þ
βj I nj ; αj I nj þ N j
j ¼ 1; : : : ; l1 ;
j ¼ l1 þ 1; : : : ; l2 ;
0 −αj I nj þ N j
j ¼ l2 þ 1; : : : ; k;
;
J r ðλkþ1 Þ ≔ N kþ1 :
3.1. The upper triangular skew-Hankel blocks form. An m × n matrix H ¼ ½hij is a Hankel matrix if hij ¼ ηiþj−1 , where fη1 ; : : : ; ηmþn−1 g are some fixed scalars; a Hankel matrix H is an upper triangular Hankel matrix if ηk ¼ 0 for all k > minfm; ng. For the matrix Dm ¼ diagð1; −1; : : : ; ð−1Þm−1 Þ, we say that an m × n matrix H is a skew-Hankel matrix if Dm H is a Hankel matrix; H is an upper triangular skew-Hankel matrix if Dm H is an upper triangular Hankel matrix. Assume there are mj Jordan blocks corresponding to the eigenvalue λj . Then the nilpotent matrix N j has the following form: ðjÞ
ðjÞ
ðjÞ
N j ¼ diagfN 1 ; N 2 ; : : : ; N mj g; ðjÞ
ðjÞ
where N i is the nilpotent block of size ni for i ¼ 1; : : : ; mj . A straightforward calculation shows that any solution Z to the matrix equation ð3:6Þ
Z N Tj þ N j Z ¼ 0
is necessarily of upper triangular skew-Hankel blocks form; that is, if we write Z into mj × mj blocks, ðjÞ
Z ¼ ½Z ik mj ×mj ;
ð3:7Þ
Z ik ∈ Rni
ðjÞ
×nk
;
then Z ik are upper triangular skew-Hankel for i; k ¼ 1; : : : ; mj . Especially, any symmetric solution Z of (3.6) is necessarily of symmetric upper triangular skew-Hankel blocks form as in (3.7) with Z ik ¼ Z Tik , where Z ii is a symmetric upper triangular skew-Hankel matrix (whose ðs; tÞ-element is zero if s þ t is odd) and Z ik ði < kÞ is an upper triangular skew-Hankel matrix.2 Any skew-symmetric solution Z of (3.6) is necessarily of skew-symmetric upper triangular skew-Hankel blocks form as in (3.7) with Z ik ¼ −Z Tik , where Z ii is a skew-symmetric upper triangular skew-Hankel matrix (whose 2
For example, 2
.
g1 60 6 Z ii ¼ 6 6 g3 40 0
0 −g3 0 0 0
g3 0 0 0 0
0 0 0 0 0
3 0 07 7 07 7; 05 0
2
Z ik
g1 6 −g2 6 ¼6 6 g3 4 −g4
g2 −g3 g4 0
g3 −g4 0 0
g4 0 0 0
3 0 07 7 07 7 05
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Z. JIA AND M. WEI
ðs; tÞ-element is equal to zero if s þ t is even)3 and Z ik ði < kÞ is an upper triangular skewHankel matrix. THEOREM 3.1. Suppose that ðX; T Þ defined by (3.1) is a real standard pair of GðλÞ described in (1.2). Then GðλÞ has a spectral decomposition (2.5), in which the parameter matrix S ∈ S T has the following structure: ð3:8Þ
S ¼ diagðS 11 ; : : : ; S l1 ;l1 ; S l1 þ1;l1 þ1 ; : : : ; S l2 ;l2 ; S l2 þ1;l2 þ1 ; : : : ; S kk ; S kþ1;kþ1 Þ;
where
ð3:9Þ ð3:10Þ ð3:11Þ
S jj
W j1 W j2 W Tj ; Wj ¼ ; j ¼ 1; : : : ; l1 ; −W j2 W j1 0 U j1 U j2 S jj ≔ ; j ¼ l1 þ 1; : : : ; l2 ; −U j2 U j1 0 V Tj2 S jj ≔ ; j ¼ l2 þ 1; : : : ; k; −V j2 0
0 ≔ −W j
W j1 , W j2 , V j2 ∈ Rnj ×nj are of upper triangular skew-Hankel blocks form; U j1 ∈ Rnj ×nj and S kþ1;kþ1 ∈ R2nkþ1 ×2nkþ1 are of skew-symmetric upper triangular skew-Hankel blocks form; U j2 ∈ Rnj ×nj are of symmetric upper triangular skew-Hankel blocks form. Proof. From Theorem 2.1, GðλÞ has a spectral decomposition (2.5) corresponding to the real standard pair ðX; T Þ defined by (3.1). Write S ¼ ðS jl Þ, where S is partitioned conforming with that of T . Then from T S ¼ ðT SÞT ¼ S T T T ¼ −ST T , we observe that ð3:12Þ
S jl J r ðλl ÞT þ J r ðλj ÞS jl ¼ 0
for j; l ¼ 1; : : : k þ 1:
We now prove that S jl ¼ 0 when j ≠ l. Equation (3.12) can be rewritten as O O I þI J r ðλj ÞvecðS jl Þ ¼ 0; ½J r ðλl Þ N in which stands for the Kronecker product N andNvec the column vectorization of a J r ðλj Þ is nonsingular; therefore, matrix. Because j ≠ l, the matrix J r ðλl Þ I þ I vecðS jl Þ ¼ 0; so S jl ¼ 0. Now we discuss the structures of the parameter matrices S jj j ¼ 1; : : : ; k þ 1, which are skew-symmetric. In the case λj ¼ αj þ iβj with αj > 0, βj > 0 and J r ðλj Þ are defined by (3.2) for j ¼ 1; : : : ; l1 ; we denote the blocks of S jj in the form 3 2 F j2 W Tj1 W Tj3 F j1 7 6 6 −F T Fj W Tj W Tj2 W Tj4 7 F~ j1 j2 7 6 ð3:13Þ S jj ≔ ¼6 7; 6 −W j1 −W j2 E j1 −W j E j E j2 7 5 4 −W j3 −W j4 −E Tj2 E~ j1 3
Here, an example of Z ii is of the form 2 0 6 6 −g2 6 6 0 6 4 −g4 0
g2 0 g4 0 0
0 −g4 0 0 0
g4 0 0 0 0
3 0 7 07 7 0 7. 7 05 0
UNDAMPED GYROSCOPIC SYSTEM
591
where F j1 , E j1 , F~ j1 , E~ j1 ∈ Rnj ×nj are skew-symmetric, F j2 , E j2 , W jk ∈ Rnj ×nj , k ¼ 1; : : : ; 4. Comparing the corresponding blocks in (3.12) with J r ðλj Þ given by (3.2), we obtain T þ F jJ þ r ðλj Þ þ J r ðλj ÞF j ¼ 0;
ð3:14Þ
T − E jJ − r ðλj Þ þ J r ðλj ÞE j ¼ 0;
T − W jJ þ r ðλj Þ þ J r ðλj ÞW j ¼ 0:
The first equation of (3.14) is equivalent to ð3:15Þ
F j1 N Tj þ N j F j1 þ 2αj F j1 þ βj ðF j2 − F Tj2 Þ ¼ 0;
ð3:16Þ
F~ j1 N Tj þ N j F~ j1 þ 2αj F~ j1 þ βj ðF Tj2 − F j2 Þ ¼ 0;
ð3:17Þ
F j2 N Tj þ N j F j2 þ 2αj F j2 þ βj ðF~ j1 − F j1 Þ ¼ 0:
Since F~ j1 and F j1 are skew-symmetric, (3.17) implies that F j2 N Tj þ N j F j2 þ 2αj F j2 is skew-symmetric, ð3:18Þ
ðF j2 þ F Tj2 ÞN Tj þ N j ðF j2 þ F Tj2 Þ þ 2αj ðF j2 þ F Tj2 Þ ¼ 0:
Equation (3.18) can be rewritten as O O ðN j I nj þ I nj N j þ 2αj I n2j ÞvecðF j2 þ F Tj2 Þ ¼ 0: ð3:19Þ Since αj ≠ 0, there must be vecðF j2 þ F Tj2 Þ ¼ 0; i.e., F Tj2 ¼ −F j2 . From (3.15) and (3.16), we obtain ðF j1 þ F~ j1 ÞN Tj þ N j ðF j1 þ F~ j1 Þ þ 2αj ðF j1 þ F~ j1 Þ ¼ 0: In a similar manner as discussing (3.18), we observe that F~ j1 ¼ −F j1 . By substituting F Tj2 ¼ −F j2 and F~ j1 ¼ −F j1 into (3.15)–(3.17), we can similarly prove F j1 ¼ 0, F~ j1 ¼ 0, and F j2 ¼ 0. In a manner similar to the above analysis, the second equation of (3.14) implies E j1 ¼ 0, E~ j1 ¼ 0, and E j2 ¼ 0, and the three equations of (3.14) implies W j1 and W j2 are of upper triangular skew-Hankel blocks forms. The proofs of the other three cases are similar to Case 1, and much simpler. Here we omit the details. ▯ Remark 3.1. For the damped nongyroscopic system, S jj are of two different forms [14]. S jj in Theorem 3.1 are of four different forms. 3.2. The semisimple structure. An eigenvalue λj is semisimple if its algebraic multiplicity nj is equal to its geometric multiplicity mj . When every eigenvalue of GðλÞ is semisimple, S is still of the form (3.8), but the upper triangular skew-Hankel structure in Theorem 3.1 will no longer show up, and S jj only retain the following properties: U j1 and S kþ1;kþ1 are skew-symmetric, and U j2 are symmetric. To further simplify the structure of S, we first recall a definition and two basic transformations in [4]. Let J ≔ ½−I0 n I0n . A matrix Q ∈ R2n×2n is called orthogonalsymplectic if and only if QJ Q T ¼ J and Q T Q ¼ I 2n (Definition 2.1 of [4]). Any Q1 Q2 , where Q 1 , orthogonal-symplectic matrix Q can be written as Q ¼ ½−Q 2 Q1
Q 2 ∈ Rn×n . Symplectic Givens rotations in R2n×2n operating in rows i, n þ i, i ∈ f1; : : : ; ng are of the form
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Z. JIA AND M. WEI
3
2 6 6 6 J s ði; θÞ ≔ 6 6 4
I i−1 cosðθÞ − sinðθÞ
7 7 7 7. 7 5
sinðθÞ I n−1 cosðθÞ I n−i
A symplectic Householder matrix is the direct sum of two Householder reflections in Rn×n ; i.e.,
Pðk; vÞ 0 P s ðk; vÞ ¼ ; 0 Pðk; vÞ
Pðk; vÞ ¼ I n − 2
vvT ; vT v
where vi ¼ 0 for i ¼ 1; : : : ; k − 1 (pp. 340–341 of [4]). Now we describe the structurepreserving reductions of two structured matrices appearing in (3.9) and (3.10). W1 W2 be a nonsingular matrix, where W 1 , W 2 ∈ Rn×n . LEMMA 3.2. Let W ¼ ½−W 2 W1 There exist two orthogonal-symplectic matrices Q 1 and Q 2 such that ð3:20Þ
Q T1 W Q 2 ¼
0 ; D
D 0
D ¼ diagðσ1 ; : : : ; σn Þ;
σi > 0:
Proof. Similar to Algorithm 4.4 in [4], we can find two orthogonal-symplectic ^ 1 and Q ^ 2 such that matrices Q 2 ð3:21Þ
^2 ¼ ^ T1 W Q Q
N
0
0
N
;
η1
6 6 6 0 N ¼6 6 6 0 4 0
3 ζ1 0 0 7 .. .. 7 . 0 7 . 7; 7 .. 0 . ζ nj−1 7 5 0
ηi ; ζ j ∈ R:
ηnj
0
There exist orthogonal matrices U , V ∈ Rnj ×nj such that N ¼ U DV T , where ^ 1 ½U 0 and Q 2 ¼ Q ^ 2 ½V 0 , we obtain D ¼ diagfσ1 ; : : : ; σnj g, σi > 0. By choosing Q 1 ¼ Q 0 U 0 V (3.20). ▯ W1 W2 Remark 3.2. We provide an example for 2n ¼ 6 to reduce W ¼ ½−W . The first 2 W1 1 step is to find a symplectic Householder matrix P s ð1; u Þ such that 2 a 6 × 6 6 6 × 1 ^ A ≔ P s ð1; u ÞW ¼ 6 6 −b 6 6 4 0 0
×
×
b
×
×
0
×
×
0
×
×
a
×
× ×
× ×3 × ×7 7 7 × ×7 7; × ×7 7 7 × ×5
×
× ×
× ×
and then find a symplectic Givens rotation J s ð1; θ1 Þ such that
593
UNDAMPED GYROSCOPIC SYSTEM
2
η1
6 6× 6 6 6× ^ ≔ J s ð1; θÞA ^ ¼6 A 60 6 6 60 4 0
× ×
0
× ×
0
× ×
0
× × η1 × ×
×
× ×
×
× ×
3
7 × ×7 7 7 × ×7 7: × ×7 7 7 × ×7 5 × ×
Following this procedure, we can finally get (3.21). U1 U2 LEMMA 3.3. Let U ¼ ½−U , where U 1 ∈ Rn×n is skew-symmetric, and U 2 ∈ Rn×n 2 U1 is symmetric. Then there exists an orthogonal-symplectic matrix Q such that
0 Q UQ ¼ −D
ð3:22Þ
T
D ; 0
in which D ¼ diagðδ1 ; : : : ; δk Þ, δi ∈ R for i ¼ 1; : : : ; k. Proof. Let J n ¼ ½−I0 n I0n , then J n U is symmetric skew-Hamiltonian. From [8], there exists an orthogonal-symplectic matrix Q such that QT J n U Q ¼
D 0
0 ; D
▯ in which D ¼ diagðδ1 ; : : : ; δn Þ, δi ∈ R. Since QJ n Q T ¼ J n , we have (3.22). Now we present the simplest possible structure of S in the semisimple case. THEOREM 3.4. Suppose that all eigenvalues of the undamped gyroscopic system GðλÞ as in (1.2) are semisimple with a real standard pair fX; T g. Then there exists a real ~ T g such that the corresponding matrix S ∈ S T has the structure standard pair fX; S ¼ diag ð3:23Þ
0 −E nl2
0 E nl1 þ1 0 I 2nl1 0 I 2n1 ; :::; ; : : : ; −I 0 ; −E nl1 þ1 0 −I 2n1 0 2nl1 0 I nl2 þ1 E nl2 0 I nkþ1 0 I nk ; ; : : : ; ; ; 0 −I nl2 þ1 0 −I nkþ1 0 −I nk 0
in which E nj ¼ diagð1; : : : ; 1Þ is of order nj × nj , j ¼ l1 þ 1; : : : ; l2 . Proof. Now we discuss the structures of S jj in four different cases mentioned at the beginning of this section. Case 1. λj ¼ αj þ iβj with αj , βj > 0, and S jj and J r ðλj Þ are defined by (3.9) and (3.2), respectively, for j ¼ 1; : : : ; l1 . Now we modify a real pair to produce the desirable W j1 W j2 structure specified in (3.23). From Lemma 3.2, for ½−W there exist orthogonalj2 W j1
symplectic matrices Q j1 , Q j2 ∈ R2nj ×2nj such that Q Tj1 ðjÞ
W j1 −W j2
ðjÞ
ðjÞ
Dj W j2 Q j2 ¼ 0 W j1
where Dj ¼ diagðσ1 ; : : : ; σnj Þ, σk > 0, k ¼ 1; : : : ; nj .
0 ; Dj
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Z. JIA AND M. WEI
Let Q j ¼ ½Q0j1
0 Qj2
for j ¼ 1; 2; : : : ; k; D jj ¼ diagðDj ; D j ; D j ; Dj Þ1∕ 2 : Then T −1 S~ jj ≔ D−1 jj Q j S jj Q j D jj ¼
I 2n1 ; 0
0 −I 2n1
and, furthermore, X~ j ¼ X j Q j D jj ; X~ j J r ðλj Þ ¼ X j Q j Djj J r ðλj Þ ¼ X j Q j J r ðλj ÞDjj ¼ X j J r ðλj ÞQ j Djj : Case 2. λj ¼ iβj with βj > 0, and S jj and J r ðλj Þ are defined by (3.10) and (3.3), respectively, for j ¼ l1 þ 1; : : : ; l2 . Here, U j1 ∈ Rnj ×nj is skew-symmetric, and U j2 ∈ Rnj ×nj is symmetric. From Lemma 3.3, S jj enjoys the special canonical form Qj1 (3.22); i.e., there exists an orthogonal-symplectic matrix Q j ¼ ½−Q j2
Q Tj S jj Q j ¼ diag ðjÞ
ðjÞ
0 −D j
Qj2 Qj1
such that
Dj ; 0
ðjÞ
in which D j ¼ diagðδ1 ; : : : ; δnj Þ, δk ∈ R for k ¼ 1; : : : ; nj . Note that Q j1 þ iQj2 is a unitary matrix in Cnj ×nj . Columns of the complex-valued n × nj matrix X~ j ≔ X j ðQ j1 þ iQ j2 ÞjDj j1∕
2
remain to represent eigenvectors of GðλÞ with corresponding eigenvalue λj . Define Djj ¼ diagðjDj j; jDj jÞ1∕ 2 . Based on (3.1), we can identify the real and the imaginary parts of X~ j as X~ j ≔ X~ jR þ iX~ jI ≔ ½ðX jR Q j1 − X jI Q j2 ÞjDj j1∕ 2 ; ðX jR Q j2 þ X jI Q j1 ÞjDj j1∕ 2 ¼ ðX jR ; X jI ÞQ j Djj : It follows that ðX~ j ; J r ðλj ÞÞ is a real pair of GðλÞ. It is easy to verify that Q Tj J r ðλj ÞQ j ¼ J r ðλj Þ, and 0 −1 T −1 ~ S jj ≔ Djj Q j S jj Q j D jj ¼ −E nl2
E nl2 0 ;
where E nj ¼ diagð1; : : : ; 1Þ is of order nj × nj . Observe further that X~ j ¼ X j Q j D jj ; X~ j J r ðλj Þ ¼ X j Q j Djj J r ðλj Þ ¼ X j Q j J r ðλj ÞDjj ¼ X j J r ðλj ÞQ j Djj : Case 3. λj ¼ αj with αj > 0, and S jj and J r ðλj Þ are defined in (3.11) and (3.4), respectively, for j ¼ l2 þ 1; : : : ; k. Denote the SVD of V j by V j ¼ Q j1 Dj Q Tj2 ;
ðjÞ
ðjÞ
D j ¼ diagðd1 ; : : : ; dnj Þ;
ðjÞ
dk > 0;
k ¼ 1; : : : ; nj ;
where Q j1 , Q j2 ∈ Rnj ×nj are orthogonal matrices. We observe that
595
UNDAMPED GYROSCOPIC SYSTEM
Q Tj S jj Q j
0 ¼ −Dj
Dj ; 0
Q j2 Qj ¼ 0
0 : Q j1
Define D jj ¼ diagðDj ; D j Þ1∕ 2 . Then 0 −1 T −1 ~ S jj ≔ Djj Q j S jj Q j D jj ¼ −I nl2
I nl 2 0 ;
and X~ j ¼ X j Q j Djj ;
X~ j J r ðλj Þ ¼ X j Q j Djj J r ðλj Þ ¼ X j Q j J r ðλj ÞDjj ¼ X j J r ðλj ÞQ j Djj :
Case 4. λkþ1 ¼ 0, J r ðλkþ1 Þ ¼ 0, and S kþ1;kþ1 is a real nonsingular skew-symmetric matrix. There exists a unitary matrix Q kþ1 ∈ R2nkþ1 ×2nkþ1 such that S kþ1;kþ1 in (3.8) has the following decomposition: 0 Dkþ1 ; Q Tkþ1 S kþ1;kþ1 Q kþ1 ¼ −Dkþ1 0 in which Dkþ1 ¼ diagðτ1 ; : : : ; τnkþ1 Þ, τk > 0, k ¼ 1; : : : ; nkþ1 . Define Dkþ1;kþ1 ¼ diagðDkþ1 ; Dkþ1 Þ1∕ 2 . Then 0 I nkþ1 −1 T −1 ~ ; S kþ1;kþ1 ≔ Dkþ1;kþ1 Q kþ1 S kþ1;kþ1 Q kþ1 Dkþ1;kþ1 ¼ −I nkþ1 0 and X~ kþ1 ¼ X kþ1 Q kþ1 Dkþ1;kþ1 ;
X~ kþ1 J r ðλkþ1 Þ ¼ X kþ1 J r ðλkþ1 ÞQ kþ1 D kþ1;kþ1 :
Now define Q ¼ diagðQ 1 ; Q 2 ; : : : ; Q k ; Q kþ1 Þ;
D ¼ diagðD11 ; D 22 ; : : : ; Dkk ; Dkþ1;kþ1 Þ1∕ 2 :
Observe that the matrix S~ ≔ D−1 Q T SQD−1 ¼ diagðS~ 11 ; S~ 22 ; : : : ; S~ kk ; S~ kþ1;kþ1 Þ has the structure specified in (3.23), and we have X~ ¼ XQD;
~ ¼ XQDT ¼ XQT D ¼ XTQD; XT
~ T Þ. therefore, S~ is indeed the matrix corresponding to the standard pair ðX; ▯ Remark 3.3. In four cases the forms of S jj are quite different, and we use different algorithms: the algorithm proposed in [4], the algorithm proposed in [8], the SVD, and the orthogonal decomposition. For the damped nongyroscopic system GðλÞ, S jj are symmetric and have two different forms. Chu and Xu [14] apply other algorithms to obtain their results. 4. Applications. In this section, we apply obtained results to solve some problems, including computations of the spectral decomposition, the QIEP, and the no spill-over eigenvalue updating problem of GðλÞ.
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Z. JIA AND M. WEI
4.1. Spectral decomposition. We now consider a two-degrees-of freedom undamped gyroscopic system containing a mass and four springs described in [32]. Example 4.1. The undamped gyroscopic system GðλÞ corresponding to the equation of motion in the rotating reference axes is
1 GðλÞ ¼ 0
0 0 2 λ þ 2Ω 1
k ∕ m − Ω2 −2Ω λþ x 0 0
0 ; ky ∕ m − Ω 2
where m is the mass, kx , ky are the stiffnesses of the springs, and Ω is the rotation rate of the system. Ω is also called gyroscopic parameter, and it characterizes the stability for this class of system. Suppose that m ¼ 1, kx ¼ 0.2, and ky ¼ 0.3; we compute the pffiffiffiffiffiffiffiffiffiffiffiffi pspecial ffiffiffiffiffiffiffiffiffiffiffiffi decomposition of GðλÞ in three different cases: 0 ≤ Ω < k ∕ m or Ω > ky ∕ m; x pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi kx ∕ m < Ω < ky ∕ m; Ω ¼ kx ∕ m or ky ∕ m. Solution. We directly give the real standard pair ðX; T Þ and the corresponding parameter matrix S in thepspectral ffiffiffiffiffiffiffiffiffiffiffiffi decomposition pffiffiffiffiffiffiffiffiffiffiffiffi (2.5). Inffiffiffiffiffiffiffiffiffiffiffiffi this case the two pairs of eigenvalues Case 1. 0 ≤ Ω < kx ∕ m or Ω > ky ∕ m. p 1 are purely imaginary. For certainty, let Ω ¼ 2 kx ∕ m ¼ 0.2236; then 1 0 0 −0.4472 0.1500 0 ð4:1Þ M¼ ; C¼ ; K¼ : 0 1 0.4472 0 0 0.2500 After some calculations we obtain 0 1.2117 0 0.9151 X¼ ; −0.8054 0 1.0665 0 3 2 0 −0.2662 0 0 7 6 7 6 0.2662 0 0 0 7 6 7 6 0 0 −0.7274 7; T ¼6 0 7 6 7 6 0 0 0.7274 0 5 4
2
0
1
6 6 −1 0 S¼6 6 0 0 4 0 0
0 0 0 −1
0
3
7 07 7: 17 5 0
pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi Case 2. kx ∕ m < Ω < ky ∕ m. This is an unstable case, since a pair of eigenvapffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi lues splits along the real axis. Let Ω ¼ ð kx ∕ m þ ky ∕ mÞ∕ 2; then −0.0475 0 0 −0.9949 1 0 : ; K¼ ; C¼ M¼ 0 0.0525 0.9949 0 0 1 After some calculations we obtain −0.7418 −0.7418 0 X¼ 0.6706 −0.6706 1.0229 2 6 0.0500 6 6 0 6 T ¼6 6 0 6 4 0
0
0
−0.0500
0
0
0
0
0.9987
0.9727 0
; 3
7 7 7 0 7 7; −0.9987 7 7 5 0 0
2
0
1
6 6 −1 0 S¼6 6 0 0 4 0
0
0 0 0 −1
0
3
7 07 7: 17 5 0
UNDAMPED GYROSCOPIC SYSTEM
597
pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi Case 3. Ω ¼ kx ∕ m or ky ∕ m. In this case zero is an eigenvalue of GðλÞ of multipffiffiffiffiffiffiffiffiffiffiffiffi plicity two. For certainty we choose Ω ¼ kx ∕ m. Then 1.0000 0 0 0.9701 X¼ ; 0 −8.9443 1.0290 0 3 3 2 2 0 −0.1111 0 0 0 1 0 0 7 7 6 6 7 6 0.1111 60 0 0 0 0 7 0 0 7 7 6 6 T ¼6 7: 7; S ¼ 6 7 7 6 0 60 0 0 0 0.9955 0 −0.9487 5 5 4 4 0 0 −0.9955 0 0 0 0.9487 0 4.2. Quadratic inverse eigenvalue problem. QIEPs have been widely studied; see [6], [10], [11], [12], [15], [17], [18], [23], [27]. We now consider a special QIEP, where the entire eigeninformation is given. Problem 4.1. Assume 2n eigenpairs fðλj ; xj Þ2n j¼1 g with • λ4j−3 ¼ λ¯ 4j−2 ¼ −λ¯ 4j−1 ¼ −λ4j ≔ αj þ βj i, αj > 0, βj > 0, x4j−3 ¼ x¯ 4j−2 ≔ xjRþ þ ixjI þ , x4j−1 ¼ x¯ 4j ≔ xjR− þ ixjI − , xjRþ , xjI þ , xjR− , xjI − ∈ Rn , j ¼ 1; 2; : : : ; l1 ; • λ2j−1 ¼ λ¯ 2j ¼ βj i, βj > 0, x2j−1 ¼ x¯ 2j ≔ xjR þ ixjI , xjR , xjI ∈ Rn , j ¼ l1 þ 1; : : : ; l2 ; • λ2j−1 ¼ −λ2j ¼ αj , αj ≥ 0, x2j−1 ≔ xjRþ , x2j ≔ xjR− , xjRþ , xjR− ∈ Rn , j ¼ l2 þ 1; : : : ; n − l1 . Find an undamped gyroscopic system GðλÞ ¼ M λ2 þ C λ þ K such that ðM λ2j þ C λj þ K Þxj ¼ 0;
ð4:2Þ
j ¼ 1; 2; : : : ; 2n:
To solve Problem 4.1, we first construct a real pair ðX; T Þ as in (3.1) from the given eigenpairs. In this2case J r ðλj Þðj ¼ 1; 2; : : :3; n − l1 Þ are defined by 6 αj 6 −βj • J r ðλj Þ ≔ 6 6 0 4 0 βj 0 ,
0 • J r ðλj Þ ≔ ½−β j
• J r ðλj Þ ≔ ½α0j
βj αj 0 0
0 −αj ,
0 0 −αj −βj
0 7 0 7 7; βj 7 5 −αj
j ¼ 1; : : : ; l1 ;
j ¼ l1 þ 1; : : : ; l2 ; j ¼ l2 þ 1; : : : ; n − l1 .
From Theorems 2.1 and 3.1, we can get a solution to Problem 4.1. THEOREM 4.1. Suppose a real pair ðX; T Þ is defined by (3.1) corresponding to given eigeninformation. Problem 4.1 has a solution GðλÞ ¼ M λ2 þ C λ þ K with M ¼ ðXT SX T Þ−1 ;
C ¼ −M XT 2 SX T M ;
K ¼ −M XT 3 SX T M þ C M −1 C ;
where the parameter matrix S ∈ S T has the structure S ¼ diagðE 1 ; : : : ; E l1 ; E l1 þ1 ; : : : ; E l2 ; E l2 þ1 ; : : : ; E n−l1 Þ; in which
598
Z. JIA AND M. WEI
2
0
0
wj1
−wj2
3
6 7 6 0 0 wj2 wj1 7 6 7 Ej ¼ 6 7; wj1 ; wj2 ∈ R; j ¼ 1; 2; : : : ; l1 ; 6 −wj1 −wj2 7 0 0 4 5 0 0 wj2 −wj1 0 ej Ej ¼ ; ej ∈ R; j ¼ l1 þ 1; : : : ; l2 ; l2 þ 1; : : : ; n − l1 −ej 0 if and only if S satisfies XSX T ¼ 0. Now we consider a QIEP described as follows. Example 4.2. Find an undamped gyroscopic system GðλÞ ¼ M λ2 þ C λ þ K which has the following eigenvalues and corresponding eigenvectors:
λ1 ¼ 0.2662i; x2 ¼
λ2 ¼ −0.2662i;
0.8328i ; −0.5535
x3 ¼
λ3 ¼ 0.7274i;
−0.6512i ; 0.7589
λ4 ¼ −0.7274i;
x4 ¼
0.6512i : 0.7589
−0.8328i x1 ¼ ; −0.5535
Solution. From above data we get a known real standard pair ðX; T Þ: X ≔ ½x1 ; x2 ; x3 ; x4 diagðP; PÞ 0 1.1778 0 ¼ −0.7828 0 1.0733
0.9209 0
;
T ≔ diagðP; PÞ diagðλ1 ; λ2 ; λ3 ; λ4 ÞdiagðP; PÞ 3 2 0 −0.2662 0 0 7 6 7 6 0.2662 0 0 0 7 6 ¼6 7; 7 6 0 0 0 −0.7274 5 4 0 0 0.7274 0 T
where P ¼ p1ffiffi2 ½11
i −i.
By Theorem 4.1 the parameter matrix S can be chosen as 2 0 6 6 −0.9884α S¼6 0 4 0
0.9884α 0 0 0
0 0 0 −0.9220α
3 0 7 0 7 7; 0.9220α 5 0
where α is a nonzero real parameter. Then we get a general solution 1 1.0710 M¼ 0 α
0 ; 1.0710
1 0 C¼ −0.4789 α
0.4789 ; 0
1 0.1606 K¼ 0 α
0 : 0.2677
UNDAMPED GYROSCOPIC SYSTEM
599
4.3. No spill-over quadratic eigenvalue updating problem. Model updating is an important tool for the design, construction, and maintenance of mechanical systems, and a number of approaches for no spill-over quadratic eigenvalue updating problems has been proposed; see [7], [9], [10], [13], [16], [25], [26]. We now consider the following no spill-over eigenvalue updating problem of the undamped gyroscopic system (1.2). Problem 4.2. Suppose that an undamped gyroscopic system GðλÞ ¼ M λ2 þ C λ þ K has only simple eigenvalues (that implies zero is not an eigenvalue of GðλÞ), and its first k ¼ 4l1 þ 2ðl2 þ l3 Þ eigenpairs fðλj ; xj Þgkj¼1 are • λ4j−3 ¼ λ¯ 4j−2 ¼ −λ¯ 4j−1 ¼ −λ4j ≔ αj þ βj i, αj > 0, βj > 0, x4j−3 ¼ x¯ 4j−2 ≔ xjRþ þ ixjI þ , x4j−1 ¼ x¯ 4j ≔ xjR− þ ixjI − , xjRþ , xjI þ , xjR− , xjI − ∈ Rn , j ¼ 1; 2; : : : ; l1 ; • λ2j−1 ¼ λ¯ 2j ¼ βj i, βj > 0, x2j−1 ¼ x¯ 2j ≔ xjR þ ixjI , xjR , xjI ∈ Rn , j ¼ l1 þ 1; : : : ; l1 þ l2 ; • λ2j−1 ¼ −λ2j ¼ αj , αj > 0, x2j−1 ≔ xjRþ , x2j ≔ xjR− , xjRþ , xjR− ∈ Rn , j ¼ l1 þ l2 þ 1; : : : ; l1 þ l2 þ l3 . ~ ~ λ2 þ C~ λ þ K~ with the Update GðλÞ to a new undamped gyroscopic system GðλÞ ¼M k k ~ first k eigenvalues fλj gj¼1 replaced by k new eigenvalues fλj gj¼1 , ¯ ¯ • λ~ 4j−3 ¼ λ~ 4j−2 ¼ −λ~ 4j−1 ¼ −λ~ 4j ≔ α~ j þ β~ j i, α~ j > 0, β~ j > 0, j ¼ 1; 2; : : : ; l~1 ; ¯ • λ~ 2j−1 ¼ λ~ 2j ¼ β~ j i, β~ j > 0, j ¼ l~1 þ 1; : : : ; l~1 þ l~2 ; • λ~ 2j−1 ¼ −λ~ 2j ¼ α~ j , α~ j > 0, j ¼ l~1 þ l~2 þ 1; : : : ; l~1 þ l~2 þ l~3 ; and the remaining 2n − k eigenvalues are unchanged. In Problem 4.2, three facts should be noted: • Complex eigenvalues with nonzero real part, purely imaginary eigenvalues, and nonzero real eigenvalues can be replaced by each other; i.e., it may happen that l1 ≠ l~1 , l2 ≠ l~2 , or l3 ≠ l~3 . • The spectrum of the updated undamped gyroscopic system still has Hamiltonian property: 4l~1 þ 2ðl~2 þ l~3 Þ ¼ 4l1 þ 2ðl2 þ l3 Þ ¼ k. ~ • The eigenvectors corresponding to the first k updated eigenvalues of GðλÞ can be freely chosen. Now we apply Theorem 2.1 to solve Problem 4.2. Suppose that ð½X 1 ; X 2 ; diagðT 1 ; T 2 ÞÞ ∈ Rn×2n × R2n×2n is a standard real pair of the origin undamped gyroscopic system GðλÞ, where ðX 1 ; T 1 Þ ∈ Rn×k × Rk×k is a real pair corresponding to the first k eigenpairs, and ðX 2 ; T 2 Þ ∈ Rn×ð2n−kÞ × Rð2n−kÞ×ð2n−kÞ is a real pair corresponding to the remaining 2n − k eigenpairs. Define X1 T C X2 T C M X1 M X2 −1 −1 SX1 ¼ ; SX2 ¼ ; X 1T 1 X 1T 1 X 2T 2 X 2T 2 −M 0 −M 0 ð4:3Þ and S ¼ diagðS X 1 ; S X 2 Þ: It is easy to show that S X 1 ∈ S T 1 , S X 2 ∈ S T 2 , and S X ∈ S T , where T ¼ diagðT 1 ; T 2 Þ. Then from Theorem 2.1 we have ð4:4Þ
X 1 S X 1 X T1 þ X 2 S X 2 X T2 ¼ 0;
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Z. JIA AND M. WEI
and M −1 ¼ X 1 T 1 S X 1 X T1 þ X 2 T 2 S X 2 X T2 ; M −1 C M −1 ¼ −X 1 T 21 S X 1 X T1 − X 2 T 22 S X 2 X T2 ; M −1 ðK − C M −1 C ÞM −1 ¼ −X 1 T 31 S X 1 X T1 − X 2 T 32 S X 2 X T2 :
ð4:5Þ
Suppose that ðX~ 1 ; T~ 1 Þ ∈ Rn×k × Rk×k is a real pair of updated undamped gyroscopic ~ system GðλÞ in Problem 4.2, where T~ 1 is corresponding to first k updated eigenvalues, ~ ~ ~ λ2 þ C~ λ þ K~ is a and X 1 is freely chosen with full column rank. Assuming GðλÞ ¼M ~ ~ solution of Problem 4.2, then ð½X 1 ; X 2 ; diagðT 1 ; T 2 ÞÞ must be one of its standard real pairs. From Theorem 2.1, we have ~ ¼ ðX~ T~ S~ X~ T Þ−1 ; M
~ X~ T~ 3 S~ X~ T M ~ þ C~ M ~ −1 C~ ; K~ ¼ −M
~ X~ T~ 2 S~ X~ T M ~; C~ ¼ −M
ð4:6Þ where X~ ¼ ½X~ 1 ; X 2 , T~ ¼ diagðT~ 1 ; T 2 Þ, and the parameter matrix S~ ¼ diagðS~ X 1 ; S X 2 Þ ∈ S ~ with S~ X ∈ S ~ . Then Problem 4.2 has a solution if and only if X~ 1 S~ X X~ T1 þ T
1
T1
1
X 2 S X 2 X T2 ¼ 0; or, equivalently, X~ 1 S~ X 1 X~ T1 ¼ X 1 S X 1 X T1 :
ð4:7Þ
n=20
n=200 15
4 original updated
3
original updated
10
2 5
1
0
0 −1
−5
−2 −10
−3 −4 −1
−0.5
0
0.5
−15 −1.5
1
−1
−0.5
0
0.5
1
1.5
n=400
20
original updated
15 10 5 0 −5 −10 −15 −20 −2
−1.5
−1
−0.5
0
0.5
1
1.5
2
FIG. 4.1. The purely imaginary eigenvalues and real eigenvalues of the original system GðλÞ are replaced by some randomly given complex eigenvalues with nonzero real part, and the complex eigenvalues with nonzero ~ real part remain invariant in the updated system GðλÞ. ◊ and þ denote the original eigenvalues and the updated eigenvalues, respectively. The case when ◊ and þ coincide means that the original eigenvalue remains invariant, i.e., no spill-over.
601
UNDAMPED GYROSCOPIC SYSTEM
Define ð4:8Þ
Z S X 1 ¼ fZ ∈ Rk×k jZ is nonsingular and Z S X 1 Z T ∈ S T~ 1 g;
X~ 1 ¼ X 1 Z −1 ;
ð4:9Þ
S~ X 1 ¼ Z S X 1 Z T ;
Z ∈ ZS X 1 :
It is clear that Z S X 1 ≠ ϕ and (4.7) holds; Problem 4.2 always has a solution. ~ ~ λ2 þ C~ λ þ K~ with THEOREM 4.2. Problem 4.2 has a solution GðλÞ ¼M ~ ¼ ½M −1 þ X 1 ðZ −1 T~ 1 Z − T 1 ÞS X X T −1 ; M 1 1 ~; ~ ½M −1 C M −1 − X 1 ðZ −1 T~ 21 Z − T 2 ÞS X X T M C~ ¼ M 1 1 1 ð4:10Þ
~ − C~ M ~ −1 C~ ; ~ ½M −1 ðK − C M −1 C ÞM −1 − X 1 ðZ −1 T~ 31 Z − T 3 ÞS X X T M K~ ¼ M 1 1 1
in which S X 1 is defined as in (4.3) and Z ∈ Z S X 1 . Now we provide two examples for the no spill-over eigenvalue updating problem. Example 4.3. Suppose that an undamped gyroscopic system GðλÞ has the following eigenvalue: λ1 ¼ 0.0500;
λ2 ¼ −0.0500;
λ3 ¼ 0.9987i;
λ4 ¼ −0.9987i;
and the corresponding eigenvectors, n=20
n=200 15
4
original updated
original updated
3
10
2 5
1
0
0 −1
−5
−2 −10
−3 −4 −0.8 −0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
−15 −1.5
−1
−0.5
0
0.5
1
1.5
n=400 20 original updated
15 10 5 0 −5 −10 −15 −20 −1.5
−1
−0.5
0
0.5
1
1.5
FIG. 4.2. The complex eigenvalues with nonzero real part and real eigenvalues of the original system GðλÞ are replaced by some randomly given purely imaginary eigenvalues, and the purely imaginary eigenvalues ~ remain invariant in the updated system GðλÞ.
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−0.7416 x1 ¼ ; 0.6708
0.7416 x2 ¼ ; 0.6708
−0.6891i x3 ¼ ; 0.7247
0.6891i x4 ¼ : 0.7247
Update the first pair of real eigenvalues to a pair of imaginary eigenvalues λ1 ¼ 0.0500i and λ2 ¼ −0.0500i; keep the remaining eigenvalues unchanged. Solution. From the given information, we get 0.0500 0 0 0.0500 T1 ¼ ; X 1 ¼ ½ x1 x2 ; T~ 1 ¼ : 0 −0.0500 −0.0500 0 Then SX1 ¼
−1.0001 : 0
0 1.0001
For any nonsingular matrix Z ∈ R2×2 we have Z S X 1 Z T ∈ S T~ 1 : So by choosing Z ¼ I 2 we ~ ~ λ2 þ C~ λ þ K~ with get a solution GðλÞ ¼M 0.0436 0 0 −0.9082 0 ~ ¼ 1.0000 : ; K~ ¼ ; C~ ¼ M 0 0.0525 0.9082 0 0 0.9174 Example 4.4. This experiment is to test the efficiency of Theorem 4.2. Suppose that GðλÞ ¼ M λ2 þ C λ þ K is a randomly given n-order undamped gyroscopic system. Respectively, update the original undamped gyroscopic system GðλÞ to a new undamped ~ gyroscopic system GðλÞ such that n=20
n=200
4
15 original updated
3
original updated
10
2 5
1 0
0
−1
−5
−2 −10
−3 −4 −1
−0.5
0
0.5
1
−15 −1.5
−1
−0.5
0
0.5
1
1.5
n=400 20 original updated
15 10 5 0 −5 −10 −15 −20 −2
−1.5
−1
−0.5
0
0.5
1
1.5
2
FIG. 4.3. The imaginary eigenvalues of the original system GðλÞ are replaced by some randomly given real ~ eigenvalues, and the real eigenvalues of GðλÞ remain invariant in the updated system GðλÞ.
UNDAMPED GYROSCOPIC SYSTEM
603
~ 1. GðλÞ has only complex eigenvalues with nonzero real part and no spill-over, ~ 2. GðλÞ has only purely imaginary eigenvalues and no spill-over, and ~ 3. GðλÞ has only real eigenvalues and no spill-over. Solution. For each subproblem, we, respectively, make three experiments in three cases that n ¼ 20, n ¼ 200, and n ¼ 400. See Figures 4.1, 4.2, and 4.3. The results demonstrate that Theorem 4.2 is efficient for solving the no spill-over quadratic eigenvalue updating problem of the undamped gyroscopic system. 5. Conclusion. In this paper we have derived a real-valued spectral decomposition of the undamped gyroscopic system GðλÞ ¼ M λ2 þ C λ þ K , in which the skew-symmetric parameter matrix S plays an important role. With a real standard pair ðX; T Þ of GðλÞ, S possesses a block diagonal structure and symmetric/skew-symmetric upper triangular skew-Hankel blocks forms. In the special case when all eigenvalues are semisimple, S has a special form as in Theorem 3.4. We have also applied these results to solve the QIEP and the no spill-over eigenvalue updating problem. Acknowledgments. The authors are grateful to Professor Peter Benner and three anonymous referees for their useful comments and suggestions, which greatly improved the original presentation.
REFERENCES [1] Z. BAI AND Y. SU, SOAR: A second-order Arnoldi method for the solution of the quadratic eigenvalue problem, SIAM J. Matrix Anal. Appl., 26 (2005), pp. 640–659. [2] M. J. BALAS, Trends in large space structure control theory: Fondest hopes, wildest dreams, IEEE Trans. Automat. Control, 27 (1982), pp. 522–535. [3] L. BARKWELL, P. LANCASTER, AND A. S. MARKUS, Gyroscopically stabilized systems: A class of quadratic eigenvalue problems with real spectrum, Canad. J. Math., 44 (1992), pp. 42–53. [4] P. BENNER, V. MEHRMANN, AND H. XU, A numerically stable, structure preserving method for computing the eigenvalues of real Hamiltonian or symplectic pencils, Numer. Math., 78 (1998), pp. 329–358. [5] A. BHAYA AND C. A. DESOER, On the design of large flexible space structures, IEEE Trans. Automat. Control, 30 (1985), pp. 1118–1120. [6] Y. CAI, Y. C. KUO, W. W. LIN, AND S. F. XU, Solutions to a quadratic inverse eigenvalue problem, Linear Algebra Appl., 430 (2009), pp. 1590–1606. [7] D. CHU, M. T. CHU, AND W. W. LIN, Quadratic model updating with symmetry, positive definiteness, and no spill-over, SIAM J. Matrix Anal. Appl., 31 (2009), pp. 546–564. [8] D. CHU, X. LIU, AND V. MEHRMANN, A numerical method for computing the Hamiltonian Schur form, Numer. Math., 105 (2007), pp. 375–412. [9] M. T. CHU, B. N. DATTA, W. W. LIN, AND S. F. XU, Spill-over phenomenon in quadratic model updating, AIAA J., 46 (2008), pp. 420–428. [10] M. T. CHU, F. DIELE, AND I. SGURA, Gradient flow methods for matrix completion with prescribed eigenvalues, Linear Algebra Appl., 379 (2004), pp. 85–112. [11] M. T. CHU AND G. H. GOLUB, Inverse Eigenvalue Problems: Theory, Algorithms, and Applications, Oxford University Press, New York, 2005. [12] M. T. CHU, Y. C. KUO, AND W. W. LIN, On inverse quadratic eigenvalue problems with partially prescribed eigenstructure, SIAM J. Matrix Anal. Appl., 25 (2004), pp. 995–1020. [13] M. T. CHU, W. W. LIN, AND S. F. XU, Updating quadratic models with no spill-over effect on unmeasured spectral data, Inverse Problems, 23 (2007), pp. 243–256. [14] M. T. CHU AND S. F. XU, Spectral decomposition of real symmetric quadratic λ-matrices and its applications, Math. Comp., 78 (2009), pp. 293–313. [15] B. N. DATTA AND V. SOKOLOV, Quadratic inverse eigenvalue problems, active vibration control and model updating, Appl. Comput. Math., 8 (2009), pp. 170–191. [16] B. N. DATTA, S. DENG, D. SARKISSIAN, AND V. SOKOLOV, An optimization technique for model updating with measured data satisfying quadratic orthogonality constraint, Mech. Syst. Signal Process., 23 (2009), pp. 1759–1772.
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[17] B. N. DATTA, S. ELHAY, Y. M. RAM, AND D. R. SARKISSIAN, Partial eigenstructure assignment for the quadratic pencil, J. Sound Vib., 230 (2000), pp. 101–110. [18] B. DONG, M. M. LIN, AND M. T. CHU, Parameter reconstruction of vibration systems from partial eigeninformation, J. Sound Vib., 327 (2009), pp. 391–401. [19] F. R. GANTMACHER, The Theory of Matrices, Vol. 1, Translated from the Russian by K. A. Hirsch, AMS Chelsea Publishing, Providence, RI, 1998. [20] I. GOHBERG, P. LANCASTER, AND L. RODMAN, Spectral analysis of self-adjoint matrix polynomials, Ann. of Math. (2), 112 (1980), pp. 33–71. [21] I. GOHBERG, P. LANCASTER, AND L. RODMAN, Matrix Polynomials, Academic Press, New York, 1982. [22] C. H. GUO, Numerical solution of a quadratic eigenvalue problem, Linear Algebra Appl., 385 (2004), pp. 391–406. [23] Y. C. KUO, W. W. LIN, AND S. F. XU, Solutions of the partially described inverse quadratic eigenvalue problem, SIAM J. Matrix Anal. Appl., 29 (2006), pp. 33–53. [24] P. LANCASTER, Lambda-Matrices and Vibrating Systems, Pergamon Press, Elmsford, NY, 1966. [25] P. LANCASTER, Model-updating for Symmetric Quadratic Eigenvalue Problems, MIMS EPrint 2006.407, Manchester Institute for Mathematical Sciences, University of Manchester, Manchester, UK (November 2006): Available online at http://eprints.ma.man.ac.uk/654. [26] P. LANCASTER, Model-updating for self-adjoint quadratic eigenvalue problems, Linear Algebra Appl., 428 (2008), pp. 2778–2790. [27] P. LANCASTER AND U. PRELLS, Inverse problems for damped vibrating systems, J. Sound Vib., 283 (2005), pp. 891–914. [28] P. LANCASTER, Isospectral vibrating systems. Part I: The spectral method, Linear Algebra Appl., 409 (2005), pp. 51–69. [29] M. M. LIN, B. DONG, AND M. T. CHU, Semi-definite programming techniques for structured quadratic inverse eigenvalue problems, Numer. Algorithms, 53 (2010), pp. 419–437. [30] J. QIAN AND W. W. LIN, A numerical method for quadratic eigenvalue problems of gyroscopic systems, J. Sound Vib., 306 (2007), pp. 284–296. [31] A. RUHE, Algorithms for the nonlinear eigenvalue problem, SIAM J. Numer. Anal., 10 (1973), pp. 674–689. [32] F. TISSEUR AND K. MEERBERGEN, The quadratic eigenvalue problem, SIAM Rev., 43 (2001), pp. 235–286.
© 2011 Society for Industrial and Applied Mathematics
A REAL-VALUED SPECTRAL DECOMPOSITION OF THE UNDAMPED GYROSCOPIC SYSTEM WITH APPLICATIONS* ZHIGANG JIA†
AND
MUSHENG WEI‡
Abstract. Given an undamped gyroscopic system GðλÞ ¼ M λ2 þ Cλ þ K with M , K symmetric and C skew-symmetric, this paper presents a real-valued spectral decomposition of GðλÞ by a real standard pair ðX; T Þ and a skew-symmetric parameter matrix S. When T is assumed to be a block diagonal matrix, the parameter matrix S has a special structure. This spectral decomposition is applied to solve the quadratic inverse eigenvalue problem and the no spill-over quadratic eigenvalue updating problem. Key words. gyroscopic system, spectral decomposition, real standard pair, skew-Hankel matrix, quadratic inverse eigenvalue problem, no spill-over eigenvalue updating problem AMS subject classifications. 15A24, 15A29, 65F18 DOI. 10.1137/100792020
1. Introduction. Gyroscopic systems in design and analysis of vibrating structures, such as bridges, highways, buildings, airplanes, etc., can be defined by a homogenous distributed parameter system (see [2] and [5], for example). Very often a distributed parameter system is first discretized to a matrix second-order model using techniques of finite element or finite difference, and then the problem is solved for this discretized reduced-order model. Associated with the matrix second-order model is the eigenvalue problem of the quadratic pencil, ð1:1Þ
QðλÞ ¼ M λ2 þ ðD þ GÞλ þ K ;
where M ¼ M T is the mass or inertia matrix, K ¼ K T the stiffness matrix, D ¼ DT the damping matrix, and G ¼ −G T the gyroscopic matrix. (Here, AT denotes the transposed matrix of A.) In many applications one often encounters two important special cases of (1.1): the damped nongyroscopic system when G ¼ 0, and the undamped gyroscopic system when D ¼ 0. The damped nongyroscopic system has been widely studied in two aspects: the quadratic eigenvalue problem (QEP), and the quadratic inverse eigenvalue problem (QIEP). The QEP analyzes and computes the spectral information; hence it reveals the dynamical behavior of the system from a priori known physical parameters such as mass, elasticity, inductance, and capacitance. The QIEP determines or estimates the parameters of the system from observed or expected eigeninformation. For the damped nongyroscopic system, Lancaster [24] and Gohberg, Lancaster, and Rodman [20], [21] studied some theoretical results of the QEP; Tisseur and Meerbergen [32] provided a good survey of applications, mathematical properties, and variety of numerical algorithms of the QEP; Bai and Su [1] proposed a second-order Arnoldi *Received by the editors April 12, 2010; accepted for publication (in revised form) by P. Benner April 4, 2011; published electronically June 29, 2011. This work was supported by the NSFC under grant 10771073 and the Shanghai Leading Academic Discipline Project under grant S30405. http://www.siam.org/journals/simax/32-2/79202.html † Department of Mathematics, Xuzhou Normal University, Jiangsu 221116, China ([email protected]). ‡ Corresponding author. College of Mathematics and Science, Shanghai Normal University, Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China ([email protected]). 584
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method for the large-scale QEP; Chu and Golub [11] provided an excellent survey of standard QIEP; Lancaster and Prells [27] discussed the QIEP with all eigenvalues being simple and nonreal from three points of view: spectral theory, structure-preserving similarity transforms, and factorization properties; Chu and Xu [14] characterized a realvalued spectral decomposition for real symmetric quadratic λ-matrices and described its applications to three challenging inverse problems: the QIEP (with the entire eigeninformation being given), the total decoupling problem, and the eigenvalue embedding problem. With only partially prescribed eigenpairs available, Chu, Kuo, and Lin [12] put forward a special solution for the symmetric QIEP. Shortly afterwards, for the QIEP with 1 < k ≤ n prescribed eigenpairs, Kuo, Lin, and Xu [23] presented a general solution and some particular solutions with additional eigenstructures. For the QIEP with n < k ≤ 2n prescribed eigenpairs, Cai et al. [6] derived a general solution in the parameterized forms in terms of two constrained parameter matrices and gave some necessary and sufficient conditions to guarantee that mass matrix is nonsingular and symmetric positive definite. Recently, Lin, Dong, and Chu [29] described an application of semidefinite programming techniques to the QIEP; Datta [15] briefly reviewed recent development of the QIEP with applications to active vibration control and finite element model updating. Study and applications of the undamped gyroscopic system are as important as the damped nongyroscopic system. The λ-matrix of the undamped gyroscopic system can be denoted by ð1:2Þ
GðλÞ ¼ M λ2 þ C λ þ K ;
where M , K , C are real square matrices with M T ¼ M and nonsingular, K T ¼ K , C T ¼ −C . It is well known that the eigenvalues of (1.2) have a Hamiltonian structure; ¯ −λ; ¯ −λÞ, possibly collapsing to real or imaginary pairs i.e., they occur in quadruples ðλ; λ; or a single zero eigenvalue. The QEP of the undamped gyroscopic system can be represented in two ways, i.e., by linearization or nonlinear matrix equations. A skew-Hamiltonian/Hamiltonian linearization of GðλÞ is M C 0 −K LðλÞ ≔ Lðλ; M ; C ; K Þ ¼ λ− : 0 M M 0 It is easy to see that GðλÞ and LðλÞ have precisely the same eigenvalues. For the eigenvalue problem of LðλÞ, Benner, Mehrmann, and Xu [4], Chu, Liu, and Mehrmann [8], and others proposed some structure-preserving algorithms by means of symplectic orthogonal transformations which perform more efficiently than the QZ and QR algorithms; see [32] for a detailed discussion. Guo [22] proposed an algorithm for finding all eigenvalues of the QEP of GðλÞ with M positive definite and K negative definite, by solving the maximal solution of a nonlinear matrix equation Z þ AT Z −1 A ¼ Q, where A ¼ M þ C þ K and Q ¼ 2ðM − K Þ. The algorithm is quadratic convergent when the QEP has no purely imaginary eigenvalues. This algorithm preserves the Hamiltonian structure of the QEP. Qian and Lin [30] considered a more general case when the QEP of GðλÞ has eigenvalues on the imaginary axis. They first computed the purely imaginary eigenvalues by the Newton’s method [31], then shifted the purely imaginary eigenvalues with no spill-over to get a new gyroscopic system which has no eigenvalues on imaginary axis, and then finally applied the method proposed by Guo [22] to the new gyroscopic system to compute the remaining eigenvalues with quadratic convergence.
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One may solve the QIEP of the undamped gyroscopic system by a powerful GLR theory developed by Gohberg, Lancaster, and Rodman [21] for general matrix polynomials of arbitrary degrees. Lancaster [28] showed that a Jordan triple ðX; J; Y Þ for a quadratic matrix polynomial can generate a unique system ðM ; C ; K Þ. In general, given a standard pair ðX; T Þ or a Jordan pair ðX; J Þ, Y can be computed by solving a nonsingular matrix equation. To obtain a self-adjoint triple ðX; J; Y Þ, Y must have a required form. In order to avoid this difficulty Lancaster [28] proposed an efficient geometric/computational approach by solving a quadratic matrix equation. Notice that the Jordan triple ðX; J ; Y Þ constructed by the GLR theory for the spectral decomposition is complex-valued. Therefore, by applying the GLR theory to a real-valued undamped gyroscopic system, one may still obtain a complex-valued spectral decomposition. Thus it becomes very interesting and challenging to derive a real-valued spectral decomposition for a real-valued undamped gyroscopic system. Chu and Xu [14] developed an elegant procedure to obtain a real-valued spectral decomposition of the damped nongyroscopic system QðλÞ ¼ M λ2 þ C λ þ K , where M , C , and K are symmetric n × n matrices and M is nonsingular. For a real standard pair ðX; ℑÞ containing the spectral data, if there exists a real nonsingular matrix S such that S T ¼ S, ℑS ¼ ðℑSÞT , and XSX T ¼ 0, then ðM ; C ; K Þ can be characterized in terms of ðX; ℑÞ and S. With a natural assumption that ℑ is block diagonal, Chu and Xu [14] derived the structure of S. In general cases, S is a symmetric block diagonal matrix with its blocks or subblocks being partitioned into upper triangular Hankel blocks; in the special case when all eigenvalues are semisimple, there exists a real standard pair such that S has a simpler structure. Inspired by the analysis in [28], [14], in this paper we derive a real-valued spectral decomposition of the undamped gyroscopic system (1.2) in terms of a real standard pair ðX; T Þ and the parameter matrix S. Here S is block diagonal satisfying S T ¼ −S, T S ¼ ðT SÞT , and XSX T ¼ 0, its blocks are of four different (symmetric/skewsymmetric) upper triangular skew-Hankel forms defined in section 3. Therefore, the parameter matrix for the undamped gyroscopic system has a more complicated formula than that for the damped nongyroscopic system discussed in [14]. If all eigenvalues of the undamped gyroscopic system are semisimple, the diagonal blocks of S are of simpler structures. But the formula of S cannot be obtained using the method of Chu and Xu [14], since the blocks of S are not symmetric Hamiltonian. Instead, we use four different algorithms: the algorithm proposed in [4], the algorithm proposed in [8], the SVD, and the orthogonal decomposition. We also apply the real-valued spectral decomposition to solve the QIEP and the no spill-over quadratic eigenvalue updating problem of the undamped gyroscopic system (1.2). Especially, we prove that the no spill-over quadratic eigenvalue updating problem of (1.2) always has a solution, and complex eigenvalues with nonzero real part, purely imaginary eigenvalues, and nonzero real eigenvalues can be replaced by each other and preserve Hamiltonian property, which are not discussed in [14]. This paper is organized as follows. In section 2, we present a real-valued spectral decomposition of a real undamped gyroscopic system. In section 3, we characterize the structure of the parameter matrix S corresponding to some special real standard pairs. In section 4, we provide some applications and examples of the spectral decomposition to the QIEP and the no spill-over quadratic eigenvalue updating problem. In section 5, we make some concluding remarks. 2. The spectral decomposition. In this section, we establish the relationship between the coefficient matrices M , C , K in (1.2), a real standard pair ðX; T Þ, and
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UNDAMPED GYROSCOPIC SYSTEM
a skew-symmetric parameter matrix S. The analysis is motivated by the work of Chu and Xu [14] for the real damped nongyroscopic system. Recall from [21] that a matrix pair ðX; T Þ ∈ Rn×2n × R2n×2n is a real standard pair X is nonsingular, and the matrix of GðλÞ if and only if the matrix U ¼ U ðX; T Þ ≔ ½XT equation ð2:1Þ
M XT 2 þ C XT þ K X ¼ 0
holds. Equation (2.1) can also be written as C M X −K ð2:2Þ T¼ −M 0 XT 0
0 −M
X : XT
It is well known that the standard pair ðX; T Þ contains precisely the same spectral information of GðλÞ. Let Ψ be a given m × m real matrix. Define ð2:3Þ
S Ψ ¼ fS ∈ Rm×m jS ¼ −S T ; ΨS ¼ ðΨSÞT g:
THEOREM 2.1. Let ðX; T Þ ∈ Rn×2n × R2n×2n with U ðX; T Þ nonsingular. Then there exists an undamped gyroscopic system GðλÞ as in (1.2) with M nonsingular such that ðX; T Þ serves as a real standard pair if and only if there exists a nonsingular matrix S ∈ S T such that ð2:4Þ
XSX T ¼ 0:
In this case, the matrix triple ðM ; C ; K Þ is given by ð2:5Þ M ¼ ðXT SX T Þ−1 ;
C ¼ −M XT 2 SX T M ;
K ¼ −M XT 3 SX T M þ C M −1 C :
Proof. Necessity. Suppose that ðX; T Þ is a real standard pair of an undamped gyroscopic system GðλÞ. Because M is real symmetrical and nonsingular and C is real C M skew-symmetrical, the matrix ½−M 0 is real skew-symmetrical and nonsingular. Define a nonsingular skew-symmetric matrix −1 X T C M X S¼ ð2:6Þ : XT −M 0 XT Equation (2.2) implies that S −1 T
¼
UT
−K 0
0 U −M
is symmetric; therefore T S is symmetric and so S ∈ S T . Then from (2.6), X X T C I 0 0 X M T SX M ¼ ; ; S ¼ XT XT 0 I I XT −M 0 from which we obtain (2.4) and the formula M in (2.5). Postmultiplying (2.1) by SX T M and using the formula of M , we obtain the formula C in (2.5). Similarly, postmultiplying (2.1) by T SX T M and using the formulas of M and C , we obtain the formula of K in (2.5).
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Sufficiency. Suppose that there exists a nonsingular matrix S ∈ S T satisfying XSX T ¼ 0. Then the matrices defined in (2.5) possess properties that M , K are symmetric and C is skew-symmetric. We then observe that
X ½ T SX T M þ SX T C XT
SX T M ¼
I 0
0 ; I
and so M XT
2
X XT
−1
¼ M XT 2 ½ T SX T M þ SX T C
SX T M ¼ ½ −K
−C ;
which is equivalent to M XT 2 þ C XT þ K X ¼ 0. ▯ Remark 2.1. It can be proven that for S ∈ S T , T 2m S is skew-symmetric and T 2mþ1 S is symmetric for a nonnegative integer m. Suppose that the assertion is true for 0 ≤ m ≤ n. Then for m ¼ n þ 1, T 2nþ2 S ¼ T ðT 2nþ1 SÞT ¼ T ð−SÞðT 2nþ1 ÞT ¼ −ðT SÞT ðT 2nþ1 ÞT ¼ −ðT 2nþ2 SÞT ; T 2nþ3 S ¼ −T ðT 2nþ2 SÞT ¼ −T ð−SÞðT 2nþ2 ÞT ¼ ðT SÞT ðT 2nþ2 ÞT ¼ ðT 2nþ3 SÞT : 3. The structure of S. In this section, for given spectral data of GðλÞ we first construct a real standard pair ðX; T Þ, where T is defined as a block diagonal matrix, and we then characterize the special structure of the parameter matrix S corresponding to ðX; T Þ. Let J ðλj Þ ¼ λj I nj þ N j be the Jordan canonical form associated with the eigenvalue λj which may contain several Jordan blocks, and let X j be the n × nj matrix whose columns form the corresponding generalized eigenspace. Here, N j is an nj × nj nilpotent matrix with ones or zeros along its superdiagonal, depending on the partial multiplicities of λj . We now separate distinct eigenvalues λj and corresponding eigentriples of GðλÞ into the following four categories. Case 1. For j ¼ 1; : : : ; l1 ; λj ¼ αj þ iβj with αj , βj > 0. Then ½X jþ ; J ðλj Þ, ½X¯ jþ ; J ðλ¯ j Þ, ½X j− ; J ð−λ¯ j Þ, ½X¯ j− ; J ð−λj Þ are eigenpairs, where X jþ ¼ X jRþ þ iX jI þ , X j− ¼ X jR− þ iX jI − , X jR , X jI ∈ Rn×nj . Case 2. For j ¼ l1 þ 1; : : : ; l2 ; λj ¼ iβj with βj > 0. Then ½X j ; J ðλj Þ, ½X¯ j ; J ðλ¯ j Þ are eigenpairs, where X j ¼ X jR þ iX jI , X jR , X jI ∈ Rn×nj . Case 3. For j ¼ l2 þ 1; : : : ; k; λj ¼ αj with αj > 0. Then ½X jþ ; J ðλj Þ, ½X j− ; J ð−λj Þ are eigenpairs, where X jþ ¼ X jRþ , X j− ¼ X jR− ∈ Rn×nj . Case 4. λkþ1 ¼ 0. Then ½X kþ1 ; J ðλkþ1 Þ is an eigenpair, where X kþ1 ∈ Rn×2nkþ1 .1 Thus, 4ðn1 þ · · · þnl1 Þ þ 2ðnl1 þ1 þ · · · þnk þ nkþ1 Þ ¼ 2n, and we can construct a real standard pair ½X; T for GðλÞ as follows: X ≔ ½X 1Rþ ; X 1I þ ; X 1R− ; X 1I − ; : : : ; X l1 Rþ ; X l1 I þ ; X l1 R− ; X l1 I − ; X ðl1 þ1ÞR ; X ðl1 þ1ÞI ; : : : ; X l2 R ; X l2 I ; X ðl2 þ1ÞRþ ; X ðl2 þ1ÞR− ; : : : ; X kRþ ; X kR− ; X kþ1 ; ð3:1Þ
T ≔ diagðJ r ðλ1 Þ; : : : ; J r ðλkþ1 ÞÞ;
where J r ðλj Þ, j ¼ 1; : : : ; k þ 1, are, respectively, defined by 1
Notice that if zero is an eigenvalue of GðλÞ, its algebra multiplicity is even.
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UNDAMPED GYROSCOPIC SYSTEM
Jþ 0 r ðλj Þ J r ðλj Þ ≔ ; 0 J− r ðλj Þ ð3:2Þ
ð3:3Þ
J r ðλj Þ ≔
J r
αj I nj þ N j ¼ −βj I nj
Nj −βj I nj
βj I nj ; Nj
αj I nj þ N j J r ðλj Þ ≔ 0
ð3:4Þ ð3:5Þ
βj I nj ; αj I nj þ N j
j ¼ 1; : : : ; l1 ;
j ¼ l1 þ 1; : : : ; l2 ;
0 −αj I nj þ N j
j ¼ l2 þ 1; : : : ; k;
;
J r ðλkþ1 Þ ≔ N kþ1 :
3.1. The upper triangular skew-Hankel blocks form. An m × n matrix H ¼ ½hij is a Hankel matrix if hij ¼ ηiþj−1 , where fη1 ; : : : ; ηmþn−1 g are some fixed scalars; a Hankel matrix H is an upper triangular Hankel matrix if ηk ¼ 0 for all k > minfm; ng. For the matrix Dm ¼ diagð1; −1; : : : ; ð−1Þm−1 Þ, we say that an m × n matrix H is a skew-Hankel matrix if Dm H is a Hankel matrix; H is an upper triangular skew-Hankel matrix if Dm H is an upper triangular Hankel matrix. Assume there are mj Jordan blocks corresponding to the eigenvalue λj . Then the nilpotent matrix N j has the following form: ðjÞ
ðjÞ
ðjÞ
N j ¼ diagfN 1 ; N 2 ; : : : ; N mj g; ðjÞ
ðjÞ
where N i is the nilpotent block of size ni for i ¼ 1; : : : ; mj . A straightforward calculation shows that any solution Z to the matrix equation ð3:6Þ
Z N Tj þ N j Z ¼ 0
is necessarily of upper triangular skew-Hankel blocks form; that is, if we write Z into mj × mj blocks, ðjÞ
Z ¼ ½Z ik mj ×mj ;
ð3:7Þ
Z ik ∈ Rni
ðjÞ
×nk
;
then Z ik are upper triangular skew-Hankel for i; k ¼ 1; : : : ; mj . Especially, any symmetric solution Z of (3.6) is necessarily of symmetric upper triangular skew-Hankel blocks form as in (3.7) with Z ik ¼ Z Tik , where Z ii is a symmetric upper triangular skew-Hankel matrix (whose ðs; tÞ-element is zero if s þ t is odd) and Z ik ði < kÞ is an upper triangular skew-Hankel matrix.2 Any skew-symmetric solution Z of (3.6) is necessarily of skew-symmetric upper triangular skew-Hankel blocks form as in (3.7) with Z ik ¼ −Z Tik , where Z ii is a skew-symmetric upper triangular skew-Hankel matrix (whose 2
For example, 2
.
g1 60 6 Z ii ¼ 6 6 g3 40 0
0 −g3 0 0 0
g3 0 0 0 0
0 0 0 0 0
3 0 07 7 07 7; 05 0
2
Z ik
g1 6 −g2 6 ¼6 6 g3 4 −g4
g2 −g3 g4 0
g3 −g4 0 0
g4 0 0 0
3 0 07 7 07 7 05
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ðs; tÞ-element is equal to zero if s þ t is even)3 and Z ik ði < kÞ is an upper triangular skewHankel matrix. THEOREM 3.1. Suppose that ðX; T Þ defined by (3.1) is a real standard pair of GðλÞ described in (1.2). Then GðλÞ has a spectral decomposition (2.5), in which the parameter matrix S ∈ S T has the following structure: ð3:8Þ
S ¼ diagðS 11 ; : : : ; S l1 ;l1 ; S l1 þ1;l1 þ1 ; : : : ; S l2 ;l2 ; S l2 þ1;l2 þ1 ; : : : ; S kk ; S kþ1;kþ1 Þ;
where
ð3:9Þ ð3:10Þ ð3:11Þ
S jj
W j1 W j2 W Tj ; Wj ¼ ; j ¼ 1; : : : ; l1 ; −W j2 W j1 0 U j1 U j2 S jj ≔ ; j ¼ l1 þ 1; : : : ; l2 ; −U j2 U j1 0 V Tj2 S jj ≔ ; j ¼ l2 þ 1; : : : ; k; −V j2 0
0 ≔ −W j
W j1 , W j2 , V j2 ∈ Rnj ×nj are of upper triangular skew-Hankel blocks form; U j1 ∈ Rnj ×nj and S kþ1;kþ1 ∈ R2nkþ1 ×2nkþ1 are of skew-symmetric upper triangular skew-Hankel blocks form; U j2 ∈ Rnj ×nj are of symmetric upper triangular skew-Hankel blocks form. Proof. From Theorem 2.1, GðλÞ has a spectral decomposition (2.5) corresponding to the real standard pair ðX; T Þ defined by (3.1). Write S ¼ ðS jl Þ, where S is partitioned conforming with that of T . Then from T S ¼ ðT SÞT ¼ S T T T ¼ −ST T , we observe that ð3:12Þ
S jl J r ðλl ÞT þ J r ðλj ÞS jl ¼ 0
for j; l ¼ 1; : : : k þ 1:
We now prove that S jl ¼ 0 when j ≠ l. Equation (3.12) can be rewritten as O O I þI J r ðλj ÞvecðS jl Þ ¼ 0; ½J r ðλl Þ N in which stands for the Kronecker product N andNvec the column vectorization of a J r ðλj Þ is nonsingular; therefore, matrix. Because j ≠ l, the matrix J r ðλl Þ I þ I vecðS jl Þ ¼ 0; so S jl ¼ 0. Now we discuss the structures of the parameter matrices S jj j ¼ 1; : : : ; k þ 1, which are skew-symmetric. In the case λj ¼ αj þ iβj with αj > 0, βj > 0 and J r ðλj Þ are defined by (3.2) for j ¼ 1; : : : ; l1 ; we denote the blocks of S jj in the form 3 2 F j2 W Tj1 W Tj3 F j1 7 6 6 −F T Fj W Tj W Tj2 W Tj4 7 F~ j1 j2 7 6 ð3:13Þ S jj ≔ ¼6 7; 6 −W j1 −W j2 E j1 −W j E j E j2 7 5 4 −W j3 −W j4 −E Tj2 E~ j1 3
Here, an example of Z ii is of the form 2 0 6 6 −g2 6 6 0 6 4 −g4 0
g2 0 g4 0 0
0 −g4 0 0 0
g4 0 0 0 0
3 0 7 07 7 0 7. 7 05 0
UNDAMPED GYROSCOPIC SYSTEM
591
where F j1 , E j1 , F~ j1 , E~ j1 ∈ Rnj ×nj are skew-symmetric, F j2 , E j2 , W jk ∈ Rnj ×nj , k ¼ 1; : : : ; 4. Comparing the corresponding blocks in (3.12) with J r ðλj Þ given by (3.2), we obtain T þ F jJ þ r ðλj Þ þ J r ðλj ÞF j ¼ 0;
ð3:14Þ
T − E jJ − r ðλj Þ þ J r ðλj ÞE j ¼ 0;
T − W jJ þ r ðλj Þ þ J r ðλj ÞW j ¼ 0:
The first equation of (3.14) is equivalent to ð3:15Þ
F j1 N Tj þ N j F j1 þ 2αj F j1 þ βj ðF j2 − F Tj2 Þ ¼ 0;
ð3:16Þ
F~ j1 N Tj þ N j F~ j1 þ 2αj F~ j1 þ βj ðF Tj2 − F j2 Þ ¼ 0;
ð3:17Þ
F j2 N Tj þ N j F j2 þ 2αj F j2 þ βj ðF~ j1 − F j1 Þ ¼ 0:
Since F~ j1 and F j1 are skew-symmetric, (3.17) implies that F j2 N Tj þ N j F j2 þ 2αj F j2 is skew-symmetric, ð3:18Þ
ðF j2 þ F Tj2 ÞN Tj þ N j ðF j2 þ F Tj2 Þ þ 2αj ðF j2 þ F Tj2 Þ ¼ 0:
Equation (3.18) can be rewritten as O O ðN j I nj þ I nj N j þ 2αj I n2j ÞvecðF j2 þ F Tj2 Þ ¼ 0: ð3:19Þ Since αj ≠ 0, there must be vecðF j2 þ F Tj2 Þ ¼ 0; i.e., F Tj2 ¼ −F j2 . From (3.15) and (3.16), we obtain ðF j1 þ F~ j1 ÞN Tj þ N j ðF j1 þ F~ j1 Þ þ 2αj ðF j1 þ F~ j1 Þ ¼ 0: In a similar manner as discussing (3.18), we observe that F~ j1 ¼ −F j1 . By substituting F Tj2 ¼ −F j2 and F~ j1 ¼ −F j1 into (3.15)–(3.17), we can similarly prove F j1 ¼ 0, F~ j1 ¼ 0, and F j2 ¼ 0. In a manner similar to the above analysis, the second equation of (3.14) implies E j1 ¼ 0, E~ j1 ¼ 0, and E j2 ¼ 0, and the three equations of (3.14) implies W j1 and W j2 are of upper triangular skew-Hankel blocks forms. The proofs of the other three cases are similar to Case 1, and much simpler. Here we omit the details. ▯ Remark 3.1. For the damped nongyroscopic system, S jj are of two different forms [14]. S jj in Theorem 3.1 are of four different forms. 3.2. The semisimple structure. An eigenvalue λj is semisimple if its algebraic multiplicity nj is equal to its geometric multiplicity mj . When every eigenvalue of GðλÞ is semisimple, S is still of the form (3.8), but the upper triangular skew-Hankel structure in Theorem 3.1 will no longer show up, and S jj only retain the following properties: U j1 and S kþ1;kþ1 are skew-symmetric, and U j2 are symmetric. To further simplify the structure of S, we first recall a definition and two basic transformations in [4]. Let J ≔ ½−I0 n I0n . A matrix Q ∈ R2n×2n is called orthogonalsymplectic if and only if QJ Q T ¼ J and Q T Q ¼ I 2n (Definition 2.1 of [4]). Any Q1 Q2 , where Q 1 , orthogonal-symplectic matrix Q can be written as Q ¼ ½−Q 2 Q1
Q 2 ∈ Rn×n . Symplectic Givens rotations in R2n×2n operating in rows i, n þ i, i ∈ f1; : : : ; ng are of the form
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3
2 6 6 6 J s ði; θÞ ≔ 6 6 4
I i−1 cosðθÞ − sinðθÞ
7 7 7 7. 7 5
sinðθÞ I n−1 cosðθÞ I n−i
A symplectic Householder matrix is the direct sum of two Householder reflections in Rn×n ; i.e.,
Pðk; vÞ 0 P s ðk; vÞ ¼ ; 0 Pðk; vÞ
Pðk; vÞ ¼ I n − 2
vvT ; vT v
where vi ¼ 0 for i ¼ 1; : : : ; k − 1 (pp. 340–341 of [4]). Now we describe the structurepreserving reductions of two structured matrices appearing in (3.9) and (3.10). W1 W2 be a nonsingular matrix, where W 1 , W 2 ∈ Rn×n . LEMMA 3.2. Let W ¼ ½−W 2 W1 There exist two orthogonal-symplectic matrices Q 1 and Q 2 such that ð3:20Þ
Q T1 W Q 2 ¼
0 ; D
D 0
D ¼ diagðσ1 ; : : : ; σn Þ;
σi > 0:
Proof. Similar to Algorithm 4.4 in [4], we can find two orthogonal-symplectic ^ 1 and Q ^ 2 such that matrices Q 2 ð3:21Þ
^2 ¼ ^ T1 W Q Q
N
0
0
N
;
η1
6 6 6 0 N ¼6 6 6 0 4 0
3 ζ1 0 0 7 .. .. 7 . 0 7 . 7; 7 .. 0 . ζ nj−1 7 5 0
ηi ; ζ j ∈ R:
ηnj
0
There exist orthogonal matrices U , V ∈ Rnj ×nj such that N ¼ U DV T , where ^ 1 ½U 0 and Q 2 ¼ Q ^ 2 ½V 0 , we obtain D ¼ diagfσ1 ; : : : ; σnj g, σi > 0. By choosing Q 1 ¼ Q 0 U 0 V (3.20). ▯ W1 W2 Remark 3.2. We provide an example for 2n ¼ 6 to reduce W ¼ ½−W . The first 2 W1 1 step is to find a symplectic Householder matrix P s ð1; u Þ such that 2 a 6 × 6 6 6 × 1 ^ A ≔ P s ð1; u ÞW ¼ 6 6 −b 6 6 4 0 0
×
×
b
×
×
0
×
×
0
×
×
a
×
× ×
× ×3 × ×7 7 7 × ×7 7; × ×7 7 7 × ×5
×
× ×
× ×
and then find a symplectic Givens rotation J s ð1; θ1 Þ such that
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UNDAMPED GYROSCOPIC SYSTEM
2
η1
6 6× 6 6 6× ^ ≔ J s ð1; θÞA ^ ¼6 A 60 6 6 60 4 0
× ×
0
× ×
0
× ×
0
× × η1 × ×
×
× ×
×
× ×
3
7 × ×7 7 7 × ×7 7: × ×7 7 7 × ×7 5 × ×
Following this procedure, we can finally get (3.21). U1 U2 LEMMA 3.3. Let U ¼ ½−U , where U 1 ∈ Rn×n is skew-symmetric, and U 2 ∈ Rn×n 2 U1 is symmetric. Then there exists an orthogonal-symplectic matrix Q such that
0 Q UQ ¼ −D
ð3:22Þ
T
D ; 0
in which D ¼ diagðδ1 ; : : : ; δk Þ, δi ∈ R for i ¼ 1; : : : ; k. Proof. Let J n ¼ ½−I0 n I0n , then J n U is symmetric skew-Hamiltonian. From [8], there exists an orthogonal-symplectic matrix Q such that QT J n U Q ¼
D 0
0 ; D
▯ in which D ¼ diagðδ1 ; : : : ; δn Þ, δi ∈ R. Since QJ n Q T ¼ J n , we have (3.22). Now we present the simplest possible structure of S in the semisimple case. THEOREM 3.4. Suppose that all eigenvalues of the undamped gyroscopic system GðλÞ as in (1.2) are semisimple with a real standard pair fX; T g. Then there exists a real ~ T g such that the corresponding matrix S ∈ S T has the structure standard pair fX; S ¼ diag ð3:23Þ
0 −E nl2
0 E nl1 þ1 0 I 2nl1 0 I 2n1 ; :::; ; : : : ; −I 0 ; −E nl1 þ1 0 −I 2n1 0 2nl1 0 I nl2 þ1 E nl2 0 I nkþ1 0 I nk ; ; : : : ; ; ; 0 −I nl2 þ1 0 −I nkþ1 0 −I nk 0
in which E nj ¼ diagð1; : : : ; 1Þ is of order nj × nj , j ¼ l1 þ 1; : : : ; l2 . Proof. Now we discuss the structures of S jj in four different cases mentioned at the beginning of this section. Case 1. λj ¼ αj þ iβj with αj , βj > 0, and S jj and J r ðλj Þ are defined by (3.9) and (3.2), respectively, for j ¼ 1; : : : ; l1 . Now we modify a real pair to produce the desirable W j1 W j2 structure specified in (3.23). From Lemma 3.2, for ½−W there exist orthogonalj2 W j1
symplectic matrices Q j1 , Q j2 ∈ R2nj ×2nj such that Q Tj1 ðjÞ
W j1 −W j2
ðjÞ
ðjÞ
Dj W j2 Q j2 ¼ 0 W j1
where Dj ¼ diagðσ1 ; : : : ; σnj Þ, σk > 0, k ¼ 1; : : : ; nj .
0 ; Dj
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Let Q j ¼ ½Q0j1
0 Qj2
for j ¼ 1; 2; : : : ; k; D jj ¼ diagðDj ; D j ; D j ; Dj Þ1∕ 2 : Then T −1 S~ jj ≔ D−1 jj Q j S jj Q j D jj ¼
I 2n1 ; 0
0 −I 2n1
and, furthermore, X~ j ¼ X j Q j D jj ; X~ j J r ðλj Þ ¼ X j Q j Djj J r ðλj Þ ¼ X j Q j J r ðλj ÞDjj ¼ X j J r ðλj ÞQ j Djj : Case 2. λj ¼ iβj with βj > 0, and S jj and J r ðλj Þ are defined by (3.10) and (3.3), respectively, for j ¼ l1 þ 1; : : : ; l2 . Here, U j1 ∈ Rnj ×nj is skew-symmetric, and U j2 ∈ Rnj ×nj is symmetric. From Lemma 3.3, S jj enjoys the special canonical form Qj1 (3.22); i.e., there exists an orthogonal-symplectic matrix Q j ¼ ½−Q j2
Q Tj S jj Q j ¼ diag ðjÞ
ðjÞ
0 −D j
Qj2 Qj1
such that
Dj ; 0
ðjÞ
in which D j ¼ diagðδ1 ; : : : ; δnj Þ, δk ∈ R for k ¼ 1; : : : ; nj . Note that Q j1 þ iQj2 is a unitary matrix in Cnj ×nj . Columns of the complex-valued n × nj matrix X~ j ≔ X j ðQ j1 þ iQ j2 ÞjDj j1∕
2
remain to represent eigenvectors of GðλÞ with corresponding eigenvalue λj . Define Djj ¼ diagðjDj j; jDj jÞ1∕ 2 . Based on (3.1), we can identify the real and the imaginary parts of X~ j as X~ j ≔ X~ jR þ iX~ jI ≔ ½ðX jR Q j1 − X jI Q j2 ÞjDj j1∕ 2 ; ðX jR Q j2 þ X jI Q j1 ÞjDj j1∕ 2 ¼ ðX jR ; X jI ÞQ j Djj : It follows that ðX~ j ; J r ðλj ÞÞ is a real pair of GðλÞ. It is easy to verify that Q Tj J r ðλj ÞQ j ¼ J r ðλj Þ, and 0 −1 T −1 ~ S jj ≔ Djj Q j S jj Q j D jj ¼ −E nl2
E nl2 0 ;
where E nj ¼ diagð1; : : : ; 1Þ is of order nj × nj . Observe further that X~ j ¼ X j Q j D jj ; X~ j J r ðλj Þ ¼ X j Q j Djj J r ðλj Þ ¼ X j Q j J r ðλj ÞDjj ¼ X j J r ðλj ÞQ j Djj : Case 3. λj ¼ αj with αj > 0, and S jj and J r ðλj Þ are defined in (3.11) and (3.4), respectively, for j ¼ l2 þ 1; : : : ; k. Denote the SVD of V j by V j ¼ Q j1 Dj Q Tj2 ;
ðjÞ
ðjÞ
D j ¼ diagðd1 ; : : : ; dnj Þ;
ðjÞ
dk > 0;
k ¼ 1; : : : ; nj ;
where Q j1 , Q j2 ∈ Rnj ×nj are orthogonal matrices. We observe that
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UNDAMPED GYROSCOPIC SYSTEM
Q Tj S jj Q j
0 ¼ −Dj
Dj ; 0
Q j2 Qj ¼ 0
0 : Q j1
Define D jj ¼ diagðDj ; D j Þ1∕ 2 . Then 0 −1 T −1 ~ S jj ≔ Djj Q j S jj Q j D jj ¼ −I nl2
I nl 2 0 ;
and X~ j ¼ X j Q j Djj ;
X~ j J r ðλj Þ ¼ X j Q j Djj J r ðλj Þ ¼ X j Q j J r ðλj ÞDjj ¼ X j J r ðλj ÞQ j Djj :
Case 4. λkþ1 ¼ 0, J r ðλkþ1 Þ ¼ 0, and S kþ1;kþ1 is a real nonsingular skew-symmetric matrix. There exists a unitary matrix Q kþ1 ∈ R2nkþ1 ×2nkþ1 such that S kþ1;kþ1 in (3.8) has the following decomposition: 0 Dkþ1 ; Q Tkþ1 S kþ1;kþ1 Q kþ1 ¼ −Dkþ1 0 in which Dkþ1 ¼ diagðτ1 ; : : : ; τnkþ1 Þ, τk > 0, k ¼ 1; : : : ; nkþ1 . Define Dkþ1;kþ1 ¼ diagðDkþ1 ; Dkþ1 Þ1∕ 2 . Then 0 I nkþ1 −1 T −1 ~ ; S kþ1;kþ1 ≔ Dkþ1;kþ1 Q kþ1 S kþ1;kþ1 Q kþ1 Dkþ1;kþ1 ¼ −I nkþ1 0 and X~ kþ1 ¼ X kþ1 Q kþ1 Dkþ1;kþ1 ;
X~ kþ1 J r ðλkþ1 Þ ¼ X kþ1 J r ðλkþ1 ÞQ kþ1 D kþ1;kþ1 :
Now define Q ¼ diagðQ 1 ; Q 2 ; : : : ; Q k ; Q kþ1 Þ;
D ¼ diagðD11 ; D 22 ; : : : ; Dkk ; Dkþ1;kþ1 Þ1∕ 2 :
Observe that the matrix S~ ≔ D−1 Q T SQD−1 ¼ diagðS~ 11 ; S~ 22 ; : : : ; S~ kk ; S~ kþ1;kþ1 Þ has the structure specified in (3.23), and we have X~ ¼ XQD;
~ ¼ XQDT ¼ XQT D ¼ XTQD; XT
~ T Þ. therefore, S~ is indeed the matrix corresponding to the standard pair ðX; ▯ Remark 3.3. In four cases the forms of S jj are quite different, and we use different algorithms: the algorithm proposed in [4], the algorithm proposed in [8], the SVD, and the orthogonal decomposition. For the damped nongyroscopic system GðλÞ, S jj are symmetric and have two different forms. Chu and Xu [14] apply other algorithms to obtain their results. 4. Applications. In this section, we apply obtained results to solve some problems, including computations of the spectral decomposition, the QIEP, and the no spill-over eigenvalue updating problem of GðλÞ.
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4.1. Spectral decomposition. We now consider a two-degrees-of freedom undamped gyroscopic system containing a mass and four springs described in [32]. Example 4.1. The undamped gyroscopic system GðλÞ corresponding to the equation of motion in the rotating reference axes is
1 GðλÞ ¼ 0
0 0 2 λ þ 2Ω 1
k ∕ m − Ω2 −2Ω λþ x 0 0
0 ; ky ∕ m − Ω 2
where m is the mass, kx , ky are the stiffnesses of the springs, and Ω is the rotation rate of the system. Ω is also called gyroscopic parameter, and it characterizes the stability for this class of system. Suppose that m ¼ 1, kx ¼ 0.2, and ky ¼ 0.3; we compute the pffiffiffiffiffiffiffiffiffiffiffiffi pspecial ffiffiffiffiffiffiffiffiffiffiffiffi decomposition of GðλÞ in three different cases: 0 ≤ Ω < k ∕ m or Ω > ky ∕ m; x pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi kx ∕ m < Ω < ky ∕ m; Ω ¼ kx ∕ m or ky ∕ m. Solution. We directly give the real standard pair ðX; T Þ and the corresponding parameter matrix S in thepspectral ffiffiffiffiffiffiffiffiffiffiffiffi decomposition pffiffiffiffiffiffiffiffiffiffiffiffi (2.5). Inffiffiffiffiffiffiffiffiffiffiffiffi this case the two pairs of eigenvalues Case 1. 0 ≤ Ω < kx ∕ m or Ω > ky ∕ m. p 1 are purely imaginary. For certainty, let Ω ¼ 2 kx ∕ m ¼ 0.2236; then 1 0 0 −0.4472 0.1500 0 ð4:1Þ M¼ ; C¼ ; K¼ : 0 1 0.4472 0 0 0.2500 After some calculations we obtain 0 1.2117 0 0.9151 X¼ ; −0.8054 0 1.0665 0 3 2 0 −0.2662 0 0 7 6 7 6 0.2662 0 0 0 7 6 7 6 0 0 −0.7274 7; T ¼6 0 7 6 7 6 0 0 0.7274 0 5 4
2
0
1
6 6 −1 0 S¼6 6 0 0 4 0 0
0 0 0 −1
0
3
7 07 7: 17 5 0
pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi Case 2. kx ∕ m < Ω < ky ∕ m. This is an unstable case, since a pair of eigenvapffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi lues splits along the real axis. Let Ω ¼ ð kx ∕ m þ ky ∕ mÞ∕ 2; then −0.0475 0 0 −0.9949 1 0 : ; K¼ ; C¼ M¼ 0 0.0525 0.9949 0 0 1 After some calculations we obtain −0.7418 −0.7418 0 X¼ 0.6706 −0.6706 1.0229 2 6 0.0500 6 6 0 6 T ¼6 6 0 6 4 0
0
0
−0.0500
0
0
0
0
0.9987
0.9727 0
; 3
7 7 7 0 7 7; −0.9987 7 7 5 0 0
2
0
1
6 6 −1 0 S¼6 6 0 0 4 0
0
0 0 0 −1
0
3
7 07 7: 17 5 0
UNDAMPED GYROSCOPIC SYSTEM
597
pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi Case 3. Ω ¼ kx ∕ m or ky ∕ m. In this case zero is an eigenvalue of GðλÞ of multipffiffiffiffiffiffiffiffiffiffiffiffi plicity two. For certainty we choose Ω ¼ kx ∕ m. Then 1.0000 0 0 0.9701 X¼ ; 0 −8.9443 1.0290 0 3 3 2 2 0 −0.1111 0 0 0 1 0 0 7 7 6 6 7 6 0.1111 60 0 0 0 0 7 0 0 7 7 6 6 T ¼6 7: 7; S ¼ 6 7 7 6 0 60 0 0 0 0.9955 0 −0.9487 5 5 4 4 0 0 −0.9955 0 0 0 0.9487 0 4.2. Quadratic inverse eigenvalue problem. QIEPs have been widely studied; see [6], [10], [11], [12], [15], [17], [18], [23], [27]. We now consider a special QIEP, where the entire eigeninformation is given. Problem 4.1. Assume 2n eigenpairs fðλj ; xj Þ2n j¼1 g with • λ4j−3 ¼ λ¯ 4j−2 ¼ −λ¯ 4j−1 ¼ −λ4j ≔ αj þ βj i, αj > 0, βj > 0, x4j−3 ¼ x¯ 4j−2 ≔ xjRþ þ ixjI þ , x4j−1 ¼ x¯ 4j ≔ xjR− þ ixjI − , xjRþ , xjI þ , xjR− , xjI − ∈ Rn , j ¼ 1; 2; : : : ; l1 ; • λ2j−1 ¼ λ¯ 2j ¼ βj i, βj > 0, x2j−1 ¼ x¯ 2j ≔ xjR þ ixjI , xjR , xjI ∈ Rn , j ¼ l1 þ 1; : : : ; l2 ; • λ2j−1 ¼ −λ2j ¼ αj , αj ≥ 0, x2j−1 ≔ xjRþ , x2j ≔ xjR− , xjRþ , xjR− ∈ Rn , j ¼ l2 þ 1; : : : ; n − l1 . Find an undamped gyroscopic system GðλÞ ¼ M λ2 þ C λ þ K such that ðM λ2j þ C λj þ K Þxj ¼ 0;
ð4:2Þ
j ¼ 1; 2; : : : ; 2n:
To solve Problem 4.1, we first construct a real pair ðX; T Þ as in (3.1) from the given eigenpairs. In this2case J r ðλj Þðj ¼ 1; 2; : : :3; n − l1 Þ are defined by 6 αj 6 −βj • J r ðλj Þ ≔ 6 6 0 4 0 βj 0 ,
0 • J r ðλj Þ ≔ ½−β j
• J r ðλj Þ ≔ ½α0j
βj αj 0 0
0 −αj ,
0 0 −αj −βj
0 7 0 7 7; βj 7 5 −αj
j ¼ 1; : : : ; l1 ;
j ¼ l1 þ 1; : : : ; l2 ; j ¼ l2 þ 1; : : : ; n − l1 .
From Theorems 2.1 and 3.1, we can get a solution to Problem 4.1. THEOREM 4.1. Suppose a real pair ðX; T Þ is defined by (3.1) corresponding to given eigeninformation. Problem 4.1 has a solution GðλÞ ¼ M λ2 þ C λ þ K with M ¼ ðXT SX T Þ−1 ;
C ¼ −M XT 2 SX T M ;
K ¼ −M XT 3 SX T M þ C M −1 C ;
where the parameter matrix S ∈ S T has the structure S ¼ diagðE 1 ; : : : ; E l1 ; E l1 þ1 ; : : : ; E l2 ; E l2 þ1 ; : : : ; E n−l1 Þ; in which
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Z. JIA AND M. WEI
2
0
0
wj1
−wj2
3
6 7 6 0 0 wj2 wj1 7 6 7 Ej ¼ 6 7; wj1 ; wj2 ∈ R; j ¼ 1; 2; : : : ; l1 ; 6 −wj1 −wj2 7 0 0 4 5 0 0 wj2 −wj1 0 ej Ej ¼ ; ej ∈ R; j ¼ l1 þ 1; : : : ; l2 ; l2 þ 1; : : : ; n − l1 −ej 0 if and only if S satisfies XSX T ¼ 0. Now we consider a QIEP described as follows. Example 4.2. Find an undamped gyroscopic system GðλÞ ¼ M λ2 þ C λ þ K which has the following eigenvalues and corresponding eigenvectors:
λ1 ¼ 0.2662i; x2 ¼
λ2 ¼ −0.2662i;
0.8328i ; −0.5535
x3 ¼
λ3 ¼ 0.7274i;
−0.6512i ; 0.7589
λ4 ¼ −0.7274i;
x4 ¼
0.6512i : 0.7589
−0.8328i x1 ¼ ; −0.5535
Solution. From above data we get a known real standard pair ðX; T Þ: X ≔ ½x1 ; x2 ; x3 ; x4 diagðP; PÞ 0 1.1778 0 ¼ −0.7828 0 1.0733
0.9209 0
;
T ≔ diagðP; PÞ diagðλ1 ; λ2 ; λ3 ; λ4 ÞdiagðP; PÞ 3 2 0 −0.2662 0 0 7 6 7 6 0.2662 0 0 0 7 6 ¼6 7; 7 6 0 0 0 −0.7274 5 4 0 0 0.7274 0 T
where P ¼ p1ffiffi2 ½11
i −i.
By Theorem 4.1 the parameter matrix S can be chosen as 2 0 6 6 −0.9884α S¼6 0 4 0
0.9884α 0 0 0
0 0 0 −0.9220α
3 0 7 0 7 7; 0.9220α 5 0
where α is a nonzero real parameter. Then we get a general solution 1 1.0710 M¼ 0 α
0 ; 1.0710
1 0 C¼ −0.4789 α
0.4789 ; 0
1 0.1606 K¼ 0 α
0 : 0.2677
UNDAMPED GYROSCOPIC SYSTEM
599
4.3. No spill-over quadratic eigenvalue updating problem. Model updating is an important tool for the design, construction, and maintenance of mechanical systems, and a number of approaches for no spill-over quadratic eigenvalue updating problems has been proposed; see [7], [9], [10], [13], [16], [25], [26]. We now consider the following no spill-over eigenvalue updating problem of the undamped gyroscopic system (1.2). Problem 4.2. Suppose that an undamped gyroscopic system GðλÞ ¼ M λ2 þ C λ þ K has only simple eigenvalues (that implies zero is not an eigenvalue of GðλÞ), and its first k ¼ 4l1 þ 2ðl2 þ l3 Þ eigenpairs fðλj ; xj Þgkj¼1 are • λ4j−3 ¼ λ¯ 4j−2 ¼ −λ¯ 4j−1 ¼ −λ4j ≔ αj þ βj i, αj > 0, βj > 0, x4j−3 ¼ x¯ 4j−2 ≔ xjRþ þ ixjI þ , x4j−1 ¼ x¯ 4j ≔ xjR− þ ixjI − , xjRþ , xjI þ , xjR− , xjI − ∈ Rn , j ¼ 1; 2; : : : ; l1 ; • λ2j−1 ¼ λ¯ 2j ¼ βj i, βj > 0, x2j−1 ¼ x¯ 2j ≔ xjR þ ixjI , xjR , xjI ∈ Rn , j ¼ l1 þ 1; : : : ; l1 þ l2 ; • λ2j−1 ¼ −λ2j ¼ αj , αj > 0, x2j−1 ≔ xjRþ , x2j ≔ xjR− , xjRþ , xjR− ∈ Rn , j ¼ l1 þ l2 þ 1; : : : ; l1 þ l2 þ l3 . ~ ~ λ2 þ C~ λ þ K~ with the Update GðλÞ to a new undamped gyroscopic system GðλÞ ¼M k k ~ first k eigenvalues fλj gj¼1 replaced by k new eigenvalues fλj gj¼1 , ¯ ¯ • λ~ 4j−3 ¼ λ~ 4j−2 ¼ −λ~ 4j−1 ¼ −λ~ 4j ≔ α~ j þ β~ j i, α~ j > 0, β~ j > 0, j ¼ 1; 2; : : : ; l~1 ; ¯ • λ~ 2j−1 ¼ λ~ 2j ¼ β~ j i, β~ j > 0, j ¼ l~1 þ 1; : : : ; l~1 þ l~2 ; • λ~ 2j−1 ¼ −λ~ 2j ¼ α~ j , α~ j > 0, j ¼ l~1 þ l~2 þ 1; : : : ; l~1 þ l~2 þ l~3 ; and the remaining 2n − k eigenvalues are unchanged. In Problem 4.2, three facts should be noted: • Complex eigenvalues with nonzero real part, purely imaginary eigenvalues, and nonzero real eigenvalues can be replaced by each other; i.e., it may happen that l1 ≠ l~1 , l2 ≠ l~2 , or l3 ≠ l~3 . • The spectrum of the updated undamped gyroscopic system still has Hamiltonian property: 4l~1 þ 2ðl~2 þ l~3 Þ ¼ 4l1 þ 2ðl2 þ l3 Þ ¼ k. ~ • The eigenvectors corresponding to the first k updated eigenvalues of GðλÞ can be freely chosen. Now we apply Theorem 2.1 to solve Problem 4.2. Suppose that ð½X 1 ; X 2 ; diagðT 1 ; T 2 ÞÞ ∈ Rn×2n × R2n×2n is a standard real pair of the origin undamped gyroscopic system GðλÞ, where ðX 1 ; T 1 Þ ∈ Rn×k × Rk×k is a real pair corresponding to the first k eigenpairs, and ðX 2 ; T 2 Þ ∈ Rn×ð2n−kÞ × Rð2n−kÞ×ð2n−kÞ is a real pair corresponding to the remaining 2n − k eigenpairs. Define X1 T C X2 T C M X1 M X2 −1 −1 SX1 ¼ ; SX2 ¼ ; X 1T 1 X 1T 1 X 2T 2 X 2T 2 −M 0 −M 0 ð4:3Þ and S ¼ diagðS X 1 ; S X 2 Þ: It is easy to show that S X 1 ∈ S T 1 , S X 2 ∈ S T 2 , and S X ∈ S T , where T ¼ diagðT 1 ; T 2 Þ. Then from Theorem 2.1 we have ð4:4Þ
X 1 S X 1 X T1 þ X 2 S X 2 X T2 ¼ 0;
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and M −1 ¼ X 1 T 1 S X 1 X T1 þ X 2 T 2 S X 2 X T2 ; M −1 C M −1 ¼ −X 1 T 21 S X 1 X T1 − X 2 T 22 S X 2 X T2 ; M −1 ðK − C M −1 C ÞM −1 ¼ −X 1 T 31 S X 1 X T1 − X 2 T 32 S X 2 X T2 :
ð4:5Þ
Suppose that ðX~ 1 ; T~ 1 Þ ∈ Rn×k × Rk×k is a real pair of updated undamped gyroscopic ~ system GðλÞ in Problem 4.2, where T~ 1 is corresponding to first k updated eigenvalues, ~ ~ ~ λ2 þ C~ λ þ K~ is a and X 1 is freely chosen with full column rank. Assuming GðλÞ ¼M ~ ~ solution of Problem 4.2, then ð½X 1 ; X 2 ; diagðT 1 ; T 2 ÞÞ must be one of its standard real pairs. From Theorem 2.1, we have ~ ¼ ðX~ T~ S~ X~ T Þ−1 ; M
~ X~ T~ 3 S~ X~ T M ~ þ C~ M ~ −1 C~ ; K~ ¼ −M
~ X~ T~ 2 S~ X~ T M ~; C~ ¼ −M
ð4:6Þ where X~ ¼ ½X~ 1 ; X 2 , T~ ¼ diagðT~ 1 ; T 2 Þ, and the parameter matrix S~ ¼ diagðS~ X 1 ; S X 2 Þ ∈ S ~ with S~ X ∈ S ~ . Then Problem 4.2 has a solution if and only if X~ 1 S~ X X~ T1 þ T
1
T1
1
X 2 S X 2 X T2 ¼ 0; or, equivalently, X~ 1 S~ X 1 X~ T1 ¼ X 1 S X 1 X T1 :
ð4:7Þ
n=20
n=200 15
4 original updated
3
original updated
10
2 5
1
0
0 −1
−5
−2 −10
−3 −4 −1
−0.5
0
0.5
−15 −1.5
1
−1
−0.5
0
0.5
1
1.5
n=400
20
original updated
15 10 5 0 −5 −10 −15 −20 −2
−1.5
−1
−0.5
0
0.5
1
1.5
2
FIG. 4.1. The purely imaginary eigenvalues and real eigenvalues of the original system GðλÞ are replaced by some randomly given complex eigenvalues with nonzero real part, and the complex eigenvalues with nonzero ~ real part remain invariant in the updated system GðλÞ. ◊ and þ denote the original eigenvalues and the updated eigenvalues, respectively. The case when ◊ and þ coincide means that the original eigenvalue remains invariant, i.e., no spill-over.
601
UNDAMPED GYROSCOPIC SYSTEM
Define ð4:8Þ
Z S X 1 ¼ fZ ∈ Rk×k jZ is nonsingular and Z S X 1 Z T ∈ S T~ 1 g;
X~ 1 ¼ X 1 Z −1 ;
ð4:9Þ
S~ X 1 ¼ Z S X 1 Z T ;
Z ∈ ZS X 1 :
It is clear that Z S X 1 ≠ ϕ and (4.7) holds; Problem 4.2 always has a solution. ~ ~ λ2 þ C~ λ þ K~ with THEOREM 4.2. Problem 4.2 has a solution GðλÞ ¼M ~ ¼ ½M −1 þ X 1 ðZ −1 T~ 1 Z − T 1 ÞS X X T −1 ; M 1 1 ~; ~ ½M −1 C M −1 − X 1 ðZ −1 T~ 21 Z − T 2 ÞS X X T M C~ ¼ M 1 1 1 ð4:10Þ
~ − C~ M ~ −1 C~ ; ~ ½M −1 ðK − C M −1 C ÞM −1 − X 1 ðZ −1 T~ 31 Z − T 3 ÞS X X T M K~ ¼ M 1 1 1
in which S X 1 is defined as in (4.3) and Z ∈ Z S X 1 . Now we provide two examples for the no spill-over eigenvalue updating problem. Example 4.3. Suppose that an undamped gyroscopic system GðλÞ has the following eigenvalue: λ1 ¼ 0.0500;
λ2 ¼ −0.0500;
λ3 ¼ 0.9987i;
λ4 ¼ −0.9987i;
and the corresponding eigenvectors, n=20
n=200 15
4
original updated
original updated
3
10
2 5
1
0
0 −1
−5
−2 −10
−3 −4 −0.8 −0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
−15 −1.5
−1
−0.5
0
0.5
1
1.5
n=400 20 original updated
15 10 5 0 −5 −10 −15 −20 −1.5
−1
−0.5
0
0.5
1
1.5
FIG. 4.2. The complex eigenvalues with nonzero real part and real eigenvalues of the original system GðλÞ are replaced by some randomly given purely imaginary eigenvalues, and the purely imaginary eigenvalues ~ remain invariant in the updated system GðλÞ.
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−0.7416 x1 ¼ ; 0.6708
0.7416 x2 ¼ ; 0.6708
−0.6891i x3 ¼ ; 0.7247
0.6891i x4 ¼ : 0.7247
Update the first pair of real eigenvalues to a pair of imaginary eigenvalues λ1 ¼ 0.0500i and λ2 ¼ −0.0500i; keep the remaining eigenvalues unchanged. Solution. From the given information, we get 0.0500 0 0 0.0500 T1 ¼ ; X 1 ¼ ½ x1 x2 ; T~ 1 ¼ : 0 −0.0500 −0.0500 0 Then SX1 ¼
−1.0001 : 0
0 1.0001
For any nonsingular matrix Z ∈ R2×2 we have Z S X 1 Z T ∈ S T~ 1 : So by choosing Z ¼ I 2 we ~ ~ λ2 þ C~ λ þ K~ with get a solution GðλÞ ¼M 0.0436 0 0 −0.9082 0 ~ ¼ 1.0000 : ; K~ ¼ ; C~ ¼ M 0 0.0525 0.9082 0 0 0.9174 Example 4.4. This experiment is to test the efficiency of Theorem 4.2. Suppose that GðλÞ ¼ M λ2 þ C λ þ K is a randomly given n-order undamped gyroscopic system. Respectively, update the original undamped gyroscopic system GðλÞ to a new undamped ~ gyroscopic system GðλÞ such that n=20
n=200
4
15 original updated
3
original updated
10
2 5
1 0
0
−1
−5
−2 −10
−3 −4 −1
−0.5
0
0.5
1
−15 −1.5
−1
−0.5
0
0.5
1
1.5
n=400 20 original updated
15 10 5 0 −5 −10 −15 −20 −2
−1.5
−1
−0.5
0
0.5
1
1.5
2
FIG. 4.3. The imaginary eigenvalues of the original system GðλÞ are replaced by some randomly given real ~ eigenvalues, and the real eigenvalues of GðλÞ remain invariant in the updated system GðλÞ.
UNDAMPED GYROSCOPIC SYSTEM
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~ 1. GðλÞ has only complex eigenvalues with nonzero real part and no spill-over, ~ 2. GðλÞ has only purely imaginary eigenvalues and no spill-over, and ~ 3. GðλÞ has only real eigenvalues and no spill-over. Solution. For each subproblem, we, respectively, make three experiments in three cases that n ¼ 20, n ¼ 200, and n ¼ 400. See Figures 4.1, 4.2, and 4.3. The results demonstrate that Theorem 4.2 is efficient for solving the no spill-over quadratic eigenvalue updating problem of the undamped gyroscopic system. 5. Conclusion. In this paper we have derived a real-valued spectral decomposition of the undamped gyroscopic system GðλÞ ¼ M λ2 þ C λ þ K , in which the skew-symmetric parameter matrix S plays an important role. With a real standard pair ðX; T Þ of GðλÞ, S possesses a block diagonal structure and symmetric/skew-symmetric upper triangular skew-Hankel blocks forms. In the special case when all eigenvalues are semisimple, S has a special form as in Theorem 3.4. We have also applied these results to solve the QIEP and the no spill-over eigenvalue updating problem. Acknowledgments. The authors are grateful to Professor Peter Benner and three anonymous referees for their useful comments and suggestions, which greatly improved the original presentation.
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