- Email: [email protected]
Diagonalizable quadratic eigenvalue problems
ARTICLE IN PRESS Mechanical Systems and Signal Processing 23 (2009) 1134–1144
Contents lists available at ScienceDirect
Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/jnlabr/ymssp
Diagonalizable quadratic eigenvalue problems Peter Lancaster a,1, Ion Zaballa b,,2 a b
Department of Mathematics and Statistics, University of Calgary, Calgary, AB, Canada T2N 1N4 ´tica Aplicada y EIO, Euskal Herriko Univertsitatea, Apdo. Correos 644, 48080 Bilbao, Spain Departamento de Matema
a r t i c l e i n f o
abstract
Article history: Received 30 May 2008 Received in revised form 3 November 2008 Accepted 6 November 2008 Available online 30 November 2008
A system is defined to be an n n matrix function LðlÞ ¼ l M þ lD þ K where M; D; K 2 Cnn and M is nonsingular. First, a careful review is made of the possibility of direct decoupling to a diagonal (real or complex) system by applying congruence or strict equivalence transformations to LðlÞ. However, the main contribution is a complete description of the much wider class of systems which can be decoupled by applying congruence or strict equivalence transformations to a linearization of a system while preserving the structure of LðlÞ. The theory is liberally illustrated with examples. & 2008 Published by Elsevier Ltd.
2
Keywords: Damped systems Decoupling Diagonalization Linearization
1. Introduction 2
A system is defined to be a matrix function Ll ¼ l M þ lD þ K where M; D; K 2 Cnn and M is nonsingular. If M; D; K 2 Rnn then we have a real system. Similarly, if M; D; K are all Hermitian, or all real and symmetric then the system is said to be Hermitian or real symmetric, respectively. The system is said to be diagonal or decoupled if M; D and K are diagonal matrices. In general M, D and K arise from the modelling of three independent physical phenomena. Two systems will be called isospectral if they share the same Jordan form; i.e. the same eigenvalues and the same partial multiplicities. Thus, a system is diagonalizable or, equivalently, it can be decoupled if it admits an isospectral diagonal system. There are some diagonalizable systems for which the diagonal form can be achieved by either a congruence transformation, LðlÞ ! U LðlÞU for some nonsingular matrix U, or a strict equivalence transformation, LðlÞ ! ULðlÞV for nonsingular matrices U and V . These are well understood but, even so, some improvement in the existing theory are made here in Section 2. Those relatively simple cases require that, in effect, one of the coefficients M, D, K is expressed in terms of the other two and their natural independence is lost. A more general diagonalization process is the main subject of this paper. The strategy is to apply certain congruence or strict equivalence transformations to the familiar isospectral ‘‘linearization’’ lA B of LðlÞ, where A¼
D
M
M
0
;
B¼
K
0
0
M
,
(1)
Corresponding author.
E-mail addresses: [email protected] (P. Lancaster), [email protected] (I. Zaballa). Partially supported by the Natural Sciences and Engineering Research Council of Canada. 2 Partially supported by the Direccio´n General de Investigacio´n Cientı´fica y Te´cnica, Proyecto de Investigacio´n MTM2007-67812-C02-01 and Gobierno Vasco GIC07/154-IT-327-07. 1
0888-3270/$ - see front matter & 2008 Published by Elsevier Ltd. doi:10.1016/j.ymssp.2008.11.007
ARTICLE IN PRESS P. Lancaster, I. Zaballa / Mechanical Systems and Signal Processing 23 (2009) 1134–1144
1135
and observe that, if M, D and K are hermitian or real symmetric, then so are A and B. This is achieved by confining attention to those transformations which preserve the block structure of A and B (the so-called structure preserving transformations). Algorithms of this kind have been studied by Garvey et al. [4,5] and by Chu and Del Buono [2,3], and it is claimed that ‘‘almost all’’ systems LðlÞ can be diagonalized in this way. Our main result is Theorem 10, in which this claim is verified and made precise in three cases: general complex or real systems and structure preserving strict equivalence, and hermitian systems and structure preserving congruence. In other words, if any system of these types is diagonalizable, then it can be diagonalized by applying the corresponding structure preserving transformation to the linearization. The key question now is what systems are diagonalizable. The answer is given in Section 3 providing a complete description of the admissible Jordan forms in such case. In particular, diagonal Jordan forms (semisimple systems) are included among them, confirming the earlier claim that ‘‘almost all’’ systems are diagonalizable. Several informative examples are included for illustration. Real application and physical implementation of these concepts are expected in the near future through the design of filters (see [6]). Our analysis depends on detailed knowledge of the theory of reduction of matrix pairs by congruence or strict equivalence transformations. For this, we rely on the recent comprehensive survey of Lancaster and Rodman [14]. However, it may be helpful to introduce some relevant ideas here. When reducing a Hermitian or real-symmetric pencil lA B by congruence without assuming that the leading coefficient is positive definite, knowledge of the notion of the sign characteristic of each real eigenvalue is required. (It appears in Sections A.3 and A.6 of the Appendix, for example, as the set of numbers Zj .) Each real eigenvalue has one or more ‘‘partial multiplicities’’ (the sizes of Jordan blocks) and a þ1 or 1 is associated with each of them. The eigenvalue has positive type if all the associated numbers are þ1, and similarly for eigenvalues of negative type. An eigenvalue has definite type if it is either of positive or negative type and, otherwise, it has mixed type. The geometric multiplicity of an eigenvalue is the number of its partial multiplicities and its algebraic multiplicity is the sum of the partial multiplicities. For completeness, and for the reader’s convenience, an Appendix is provided giving a summary of canonical forms for linearizations under either congruence or strict equivalence transformations. 2. Diagonalization without linearization There are many engineering applications in which the system is Hermitian and M is positive definite (M40) and, in the absence of a good mathematical model of the damping phenomenon, D is supposed to be a linear combination of M and K (the hypothesis of ‘‘proportional damping’’). Some straightforward generalizations arise frequently in the engineering and computational literature and are variously known as ‘‘modal’’ or ‘‘Rayleigh’’ damping. See [17], for example, for an interesting recent study. Several variations on the same theme can arise according as strict equivalence or congruence transformations are used, the coefficients are real or complex, and with or without symmetry. But in every case, if three matrices are to be diagonalized simultaneously then the Caughey–O’Kelly commutativity condition DM 1 K ¼ KM 1 D, or something equivalent, is required (see [1,15,16]). 2.1. Hermitian systems: reduction by congruence When a system has Hermitian coefficients and the condition M40 is relaxed (as below), then real eigenvalues can arise and, as described above, they have either positive type, negative type, or mixed type. (See [14] or Appendix B of [11], for example.) Here, we first admit semisimple (i.e. nondefective) real eigenvalues with no restriction on the type. Lemma 1. Let M; K 2 Cnn with det Ma0, M ¼ M, K ¼ K. Assume that lM þ K is semisimple with all eigenvalues real and define
L ¼ diag½l1 I 1 ; l2 I 2 ; . . . ; ls I s ;
S ¼ diag½I 1 ; I 2 ; . . . ; I s ,
(2)
where the size of the identity matrix I j is a partial multiplicity of eigenvalue lj for each j, and the sign of each term in S is determined by the corresponding þ1 or 1 in the sign characteristic. Then there exists a family of nonsingular matrices V 2 Cnn such that V MV ¼ S;
V KV ¼ S L.
(3)
If V is one such matrix, then so is any matrix VA where A ¼ diag½A1 ; A2 ; . . . ; As and each Aj is unitary with the size of I j . Proof. This is a classical result. It is a special case of Theorem 6.1 of [14]. (Note that there may be repetitions among the lj .) & A result closely analogous to Lemma 1 holds in the case that M; K 2 Rnn . It is only necessary to use congruence over R and to replace the unitary matrices Aj by real orthogonal matrices. (This is a special case of Theorem 9.2 of [14].) Note carefully that the further condition M40 would ensure that all eigenvalues are real and of positive type,
ARTICLE IN PRESS 1136
P. Lancaster, I. Zaballa / Mechanical Systems and Signal Processing 23 (2009) 1134–1144
and then S ¼ I. This is the case in the original paper of Caughey and O’Kelly of 1965 [1]. Their theorem can be generalized as follows: Theorem 2. Let the hypotheses of Lemma 1 hold, assume also that all eigenvalues of lM þ K have definite type, and that D ¼ D. Then there exists a nonsingular U 2 Cnn such that U MU, U DU, and U KU are diagonal if and only if DM 1 K ¼ KM 1 D. If, in addition, M; D; K are real and symmetric, then there is a corresponding U 2 Rnn . Proof. We use the notations introduced in Lemma 1. From (3) we obtain M 1 K ¼ ðVSV ÞðV S LV 1 Þ ¼ VS 2 LV 1 ¼ V LV 1 ,
(4)
KM 1 ¼ ðV S LV 1 ÞðVSV Þ ¼ V S LSV ¼ V LU .
(5)
If also DM
1
K ¼ KM
1
D then DV LV
1
¼V
LV D and so
ðV DV ÞL ¼ LðV DV Þ. The assumption that all eigenvalues have definite type means that the eigenvalues l1 ; . . . ; ls of (2) are distinct. And so it follows that V DV is block-diagonal (as in (2)). But, as the blocks of V DV are Hermitian, the unitary blocks of matrix A of Lemma 1 can be chosen to further reduce V DV to diagonal form. Conversely, if M 0 :¼U MU; D0 :¼U DU; K 0 ¼ U KU are diagonal, it is easily verified that DM 1 K ¼ KM 1 D. The case of real-symmetric matrices M; D; K is very similar and depends on the analogue of Lemma 1 mentioned above. & 2.2. No symmetry: reduction by strict equivalence For systems which are not symmetric (or Hermitian) it is natural to replace the congruence transformations of LðlÞ used above by strict equivalence transformations. This possibility was investigated by Ma and Caughey [16] and it is interesting that, in this case too, the commutativity condition of Caughey–O’Kelly continues to play a restrictive role. However, Theorem 3 of [16] is false as it stands—a stronger hypothesis is required. A counter-example for that theorem is 0 1 1 0 2 1 0 LðlÞ ¼ l þl þ . 0 1 0 0 0 1 One possibility is to replace the condition that lM þ K is semisimple by requiring distinct eigenvalues for this pencil (which is obviously not satisfied by the counter-example). Lemma 3. Let M; K 2 Cnn with det Ma0, assume that lM þ K is semisimple, and write a diagonal matrix of the eigenvalues of lM þ K in the form
L ¼ diag½l1 I 1 ; l2 I 2 ; . . . ; ls I s , where li alj when iaj. Then there is a family of nonsingular matrices U; V 2 Cnn such that UMV ¼ I
and
UKV ¼ L.
(6) 1
If A ¼ diag½A1 ; . . . ; As is nonsingular and Aj has the size of I j , then U; V can be replaced by A
U; VA; respectively.
Proof. This is the classical result known as reduction to Kronecker form. It is a special case of Theorem 3.1 of [14].
&
Once again, there is a real analogue of this result. The reader will be able to formulate this analogue (for real strict equivalence transformations of real systems) using Theorem 3.2 of [14], for example. Theorem 4. Let M; D; K 2 Cnn with det Ma0 and assume that lM þ K has n distinct eigenvalues. Then there exist nonsingular U; V 2 Cnn such that UMV ¼ I, and UDV , UKV are diagonal if and only if DM 1 K ¼ KM 1 D. Proof. First use Lemma 3 to obtain nonsingular U; V 2 Cnn such that UMV ¼ I and UKV ¼ L, a diagonal matrix. Then M 1 K ¼ ðVUÞðU 1 LV 1 Þ ¼ V LV 1 ,
(7)
KM 1 ¼ ðU 1 LV 1 ÞðVUÞ ¼ U 1 LU. If also DM
1
K ¼ KM
1
D then DðV LV
1
(8) Þ ¼ ðU
1
LUÞD and hence (9)
ðUDV ÞL ¼ LðUDV Þ. Since L is diagonal with distinct diagonal entries, this implies that UDV is also diagonal, as required. Conversely, if M 0 :¼UMV ; D0 :¼UDV ; K 0 ¼ UKV are diagonal, it is easily verified that DM 1 K ¼ KM 1 D.
&
ARTICLE IN PRESS P. Lancaster, I. Zaballa / Mechanical Systems and Signal Processing 23 (2009) 1134–1144
1137
If the assumption that lM þ K has distinct eigenvalues is relaxed and we assume only that lM þ K is semisimple (as in [16]), then the proof of the last theorem can be followed as far as Eq. (9). One may then be able to use the freedom of choice of matrix A of Lemma 3 to reduce the blocks of UDV by similarity (the strategy proposed in [16]). This will be possible provided that UDV (and hence each diagonal block of UDV ) is semisimple. 3. Jordan forms for diagonal systems As shown in the previous section, most systems cannot be diagonalized by strict equivalence or congruence; a very strong commutativity condition must be satisfied. However, as we see now, most systems admit diagonal isospectral systems; i.e. diagonal systems with the same Jordan form as the original one. As in the Introduction, these systems will be called diagonalizable and we aim to completely describe them through their Jordan forms. Once a system is known to be diagonalizable, the natural question is how to reduce it to diagonal form. It is shown in Section 5 that this can always be achieved by means of structure preserving transformations applied to the linearizations lA B given in (1). In order to describe the Jordan forms of diagonalizable systems, we consider the primitive case of systems where the coefficients M; D; K are already diagonal. They can be seen as the ‘‘target’’ systems to which more general systems are to be reduced (whenever possible). The Jordan form for a system LðlÞ is the classical canonical form for A1 B under similarity transformations over C (used here unless specified otherwise), or over R. Definition 5. Let Jn;C and Jn;R be the classes of 2n 2n canonical Jordan matrices for n n diagonal systems, and n n real diagonal systems, respectively (so that Jn;R Jn;C C2n2n ). A Jordan block with eigenvalue lj and size s is denoted by 2 3 lj 1 0 0 60 l 1 0 7 j 6 7 6 7 6 .. .. 7 . 7 J s ðlj Þ:¼6 6 . 7. 6 . 7 6 .. 0 17 4 5 0 0 0 lj L It will be convenient to denote a direct (diagonal) sum of scalars or matrices, x1 ; . . . ; xk by kj¼1 xj. It is clear that, given any set of 2n complex numbers, they can be sorted into n pairs and n scalar quadratics are determined, each having one pair of the given numbers as its zeros. Diagonal Jordan matrices in the class Jn;C can be formed in this way. However, a similar construction for Jn;R requires that the 2n numbers consist of, say, r pairs of real numbers (0prpn) and n r pairs of (nonreal) conjugate complex numbers. As long as all 2n numbers are distinct the corresponding Jordan forms will be diagonal, but the situation becomes more complicated if repetitions among the eigenvalues are permitted. Assume that there exist distinct eigenvalues l1 ; . . . ; lt 2 C; 1ptp2n, and let li have partial multiplicities ni1 X Xni;mg;i 40 for each i (forming the ‘‘Segre characteristic’’). Then eigenvalue li has geometric multiplicity mg;i pn Pmg;i and algebraic multiplicity ma;i ¼ j¼1 nij p2n. Also, mg;i t X X
nij ¼ 2n.
(10)
i¼1 j¼1
L Q 2 Write LðlÞ ¼ ni¼1 ½mi l þ di l þ ki ; where ni¼1 mi a0. Then each diagonal entry has a linearization " # 0 1 lI 2 ; i ¼ 1; 2; . . . ; t, ki =mi di =mi and LðlÞ has the tridiagonal linearization lI A where " # n M 0 1 A¼ . ki =mi di =mi
(11)
(12)
i¼1
Furthermore, the elementary divisors of lI A are just the disjoint union of those of (11). Hence, 1pnij p2
for 1pipt; 1pjpmg;i .
For each distinct eigenvalue li , i.e. for i ¼ 1; 2; . . . ; t, we define the integers si X0 by writing ( 2 for j ¼ 1; 2; . . . ; si ; nij ¼ 1 for j ¼ si þ 1; . . . ; mg;i :
(13)
(14)
Each partial multiplicity nij ¼ 2 is necessarily associated with just one block of (12) and so, if p ¼ s1 þ s2 þ þ st (the number of quadratic elementary divisors) then the remaining n p diagonal blocks of A cannot have repeated eigenvalues
ARTICLE IN PRESS 1138
P. Lancaster, I. Zaballa / Mechanical Systems and Signal Processing 23 (2009) 1134–1144
because they are nonderogatory matrices. Thus, in (14) we have
mg;i si pn p;
i ¼ 1; 2; . . . ; t.
(15)
We now claim: i¼t;j¼m
Theorem 6. A Jordan matrix J with Segre characteristic fnij gi¼1;j¼1g;i (defined as above) is in Jn;C if and only if conditions (10), (13) and (15) hold where, for i ¼ 1; 2; . . . ; t the integers si X0 appearing in (15) are defined by (14). Proof. The necessity of the three conditions has been established. Suppose, conversely, that a Jordan matrix satisfies (10), (13) and (15). As above, the distinct eigenvalues of J are l1 ; . . . ; lt with geometric multiplicities mg;1 ; . . . ; mg;t , respectively, and s1 ; . . . ; st are the respective numbers of partial multiplicities equal to 2. Furthermore, it may be assumed that
mg;1 s1 Xmg;2 s2 X Xmg;t st . There are second degree elementary divisors for i ¼ 1; 2; . . . ; t and j ¼ 1; 2; . . . ; si . Let Aij be their corresponding 2 2 companion matrices. The remaining elementary divisors are linear and their eigenvalues can be sorted into distinct pairs, say ðlki ; lkj Þ with lki alkj . Since the number of elementary divisors of degree two is p:¼s1 þ s2 þ þ st , the number of linear elementary P divisors is 2ðn pÞ ¼ ti¼1 ðmg;i si Þ. Conditions (15) and (10) ensure that the corresponding eigenvalues can be sorted into n p pairs ðlki ; lkj Þ with lki alkj . In other words, the eigenvalues with linear elementary divisors can be organized in two ordered lists, each with n p eigenvalues, say and
ðapþ1 ; apþ2 ; . . . ; an Þ
ðbpþ1 ; bpþ2 ; . . . ; bn Þ
with ai abi for i ¼ p þ 1; . . . ; n. For each such i let Bi be the companion matrix of ðl ai Þðl bi Þ and observe that each Bi is diagonalizable. Now the matrix A ¼ diagðA11 ; . . . ; A1s1 ; . . . ; At1 ; . . . ; Atst ; Bpþ1 ; . . . ; Bn Þ has J for its Jordan form and, also, lI 2n A is the linearization of a (monic) diagonal matrix polynomial of degree 2, i.e. J 2 Jn;C . & It follows immediately from (13) that a necessary, but not sufficient condition that J 2 Jn;C is
ma;i p2mg;i for i ¼ 1; 2; . . . ; t.
(16)
Each one of the following examples shows a particular feature related to the possibility of diagonalizing the given system. Example 1. Let 2 1 LðlÞ ¼ l 0
0 2 þl 1 1
1 1 þ 2 1
1 . 2
It is easily seen that det LðlÞ ¼ ðl þ 1Þ4 so there is just one eigenvalue, 1, with ma ¼ 4. Also, Lð1Þ ¼ ½00 10, so mg ¼ 1. It follows that J consists of just one Jordan block of size four. Conditions (13) and (16) above are not satisfied, so JeJn;C and the system is not diagonalizable. The commutativity condition DM 1 K ¼ KM 1 D may or may not hold (in this case it does not hold but there are nondiagonalizable systems for which this condition is satisfied, see Example 2), so Theorem 2 does not apply. Example 2. Let 2
3
2
1
1
1
1
26 LðlÞ ¼ l 4 1 1
0 1
6 7 1 5 þ l4 1 3=2 0 2
1 0 2
3=2
3
2
0
6 2 7 5þ4 0 0 1=2
4
0 0 1
1=2
3
1 7 5. 0
In this case det LðlÞ ¼ l ðl 1Þ . The eigenvalue þ1 has ma ¼ 4 and the rank of Lð1Þ is 0. Thus mg ¼ 3 and condition (16) is 2 fulfilled. For this system, however, JeJn;C because the elementary divisors of LðlÞ are ðl 1Þ2 , ðl 1Þ, ðl 1Þ and l and condition (15) is not satisfied. Although DM 1 K ¼ KM 1 D in this case, LðlÞ cannot be decoupled by strict equivalence; otherwise, it would have a Jordan form satisfying condition (15). Also, Theorems 2 and 4 do not apply because M 1 K has the defective eigenvalue zero with algebraic multiplicity ma ¼ 2 and geometric multiplicity mg ¼ 1. Example 3. The following two systems: 73 36 32 2 41 12 L1 ðlÞ ¼ l þl þ 12 34 36 52 24
24 18
ARTICLE IN PRESS P. Lancaster, I. Zaballa / Mechanical Systems and Signal Processing 23 (2009) 1134–1144
1139
and 2
L2 ðlÞ ¼ l
0 1
1 þl 3 3
1
3 7
þ
1
2
2
4
have the same Jordan structure. In fact, det L1 ðlÞ and det L2 ðlÞ are scalar multiple of lðl 1Þ3 . So, for the eigenvalue þ1, ma ¼ 3 and since L1 ð1Þ ¼ L2 ð1Þ ¼ 0, this eigenvalue has geometric multiplicity mg ¼ 2. Actually, for both systems the elementary divisors are: ðl 1Þ2 , ðl 1Þ and l. Thus, all conditions of Theorem 6 are satisfied and L1 ðlÞ, L2 ðlÞ and the diagonal system 1 0 1 0 0 0 ^ lÞ ¼ l2 Lð þl þ 0 1 0 2 0 1 are isospectral. However, there is an important difference between L1 ðlÞ and L2 ðlÞ: the mass matrix in L1 ðlÞ is positive definite and in ^ lÞ are ‘‘strictly’’ isospectral, in the sense of [13] (the sign characteristics are the L2 ðlÞ it is not. It turns out that L1 ðlÞ and Lð ^ same), and L2 ðlÞ and LðlÞ are not strictly isospectral. ^ lÞ are In addition, for all three systems DM 1 K ¼ KM 1 D. In particular, Theorem 2 applies for L1 ðlÞ; i.e. L1 ðlÞ and Lð congruent. Also, Theorem 4 applies for L2 ðlÞ. That is to say, L2 ðlÞ can be reduced to monic diagonal form by real strict equivalence. Example 4. Let 5 2 LðlÞ ¼ l 2
2
1
319 þl 126 2
2666 1053
þ
0
319
0
126
.
2
For this system det LðlÞ ¼ l ðl 1Þ. For the eigenvalue 0, ma ¼ 2 and mg ¼ 1 and for the eigenvalues þ1 and 1, ma ¼ mg ¼ 1. Thus all conditions of Theorem 6 hold and an isospectral diagonal system is, for example, 2 0 0 0 ^ lÞ ¼ l2 Lð þ . 0 1 0 1 However, although lM þ K has distinct eigenvalues, DM 1 KaKM 1 D and so,by Theorem 4, LðlÞ cannot be decoupled by strict equivalence. 4. Jordan forms for real diagonal systems Now consider Jordan canonical forms for real diagonal systems. i¼t;j¼m
Theorem 7. A Jordan matrix J with Segre characteristic fnij gi¼1;j¼1g;i (defined as above) is in Jn;R if and only if there is an n0 , 0pn0 pn, such that J ¼ diagðJ n0 ; J nn0 Þ for Jordan matrices J n0 ; J nno with sðJ n0 Þ R and sðJ nn0 Þ \ R ¼ ; and: (a) conditions (10), (13) and (15) (with n replaced by n0 ) hold for J n0 and (b) sðJ nn0 Þ consists of conjugate pairs of nonreal semisimple eigenvalues lj ; lj . Note that, if the system (and hence J) has no real eigenvalues then n0 ¼ 0 and J n0 simply does not appear, and if the system (and hence J) has no nonreal eigenvalues then n0 ¼ n and J nn0 does not appear. To illustrate, in the case of Example 4, n0 ¼ n and so J nn0 does not appear and 2 3 0 1 0 0 60 0 0 0 7 6 7 J ¼ J n0 ¼ 6 7. 40 0 1 0 5 0
0
0
1
Proof. If LðlÞ is a real diagonal system and l0 is an eigenvalue with l0 al0 , then unit co-ordinate vectors ej can be chosen as corresponding eigenvectors, and the number of such independent vectors is just the algebraic multiplicity of l0 . Thus, l0 is semisimple, and similarly for all nonreal eigenvalues. After permutation, the set of all nonreal eigenvalues determines the matrix J nn0 satisfying condition (b). For the complementary part of J all eigenvalues are real and the argument used in the proof of Theorem 6 can be applied to obtain condition (a). (The argument over the real field is the same as that over the complex numbers used in the earlier proof.) The sufficiency of the prescribed conditions is clear. &
ARTICLE IN PRESS 1140
P. Lancaster, I. Zaballa / Mechanical Systems and Signal Processing 23 (2009) 1134–1144
Systems with all eigenvalues semisimple clearly satisfy the conditions of Theorem 6, if complex, or Theorem 7, if real. So, we have Corollary 8. If all eigenvalues of LðlÞ are semisimple and J is its Jordan form then J 2 Jn;C if LðlÞ is a complex system and J 2 Jn;R if LðlÞ is real. Example 5 (See Lancaster [10, Example 13.1]). This example shows that, in contrast to condition (b) above, a real-symmetric system may have nonreal eigenvalues with nonlinear elementary divisors. Thus, " pffiffiffi # 1 3=2 1 0 2 1 0 l þ pffiffiffi lþ LðlÞ:¼ (17) 3=2 2 0 1 0 4 pffiffiffiffiffiffi has eigenvalues 14ð3 i 23Þ with elementary divisors of degree two. Example 6. The following system, however, satisfies the conditions of Theorem 7: " # " # 3=2 1=2 2 11=2 9=2 3 5 l þ lþ . LðlÞ ¼ 1=2 3=2 9=2 11=2 5 3 In fact, its eigenvalues are: l1 ¼ 1, l2 (i.e. ma ¼ 2 and mg ¼ 1), its Jordan form is 1 0 2 0 1 ^ lÞ ¼ l2 Lð þl þ 0 1 0 4 0
¼ 2 þ i and l3 ¼ 2 i. Since the eigenvalue 1 has partial multiplicity (2) in Jn;R . A diagonal isospectral system is 0 . 5
Furthermore, it is easily verified that DM 1 K ¼ KM 1 D. Since M is positive definite, Theorem 2 applies and LðlÞ can be decoupled by congruence (applied to LðlÞ itself) to obtain a real diagonal system. Example 7. The final example corresponds to a simple two-degree-of-freedom mass–spring system: 200 100 2000 1000 2 1 0 LðlÞ ¼ l þl þ . 0 2 100 100 1000 2000 This is not a proportionally damped system and Theorems 2 and 4 show that LðlÞ cannot be reduced to diagonal form by either congruence or strict equivalence. However, the eigenvalues of LðlÞ are all distinct: 217:2699, 10:5432, 11:0935 þ 23:0598i, 11:0935 23:0598i. By Corollary 8, the Jordan form of LðlÞ is in J n;R . A diagonal system with the same Jordan form is 227:8131 0 2290:7119 0 1 0 ^ lÞ ¼ l2 Lð þ þl . 0 22:1869 0 654:8182 0 1
5. Linearization and diagonalizable systems Given a system LðlÞ, we now consider the generation of an isospectral diagonal system (when one exists) by the application of strict equivalence, or congruence transformations to the linearization lA B (see (1)). The following specific classes of transformations will be considered. Notice that they are all ‘‘structure preserving’’ transformations in the sense that the block structures of (1) are preserved and, in each case, the transformations are defined over the complex numbers. Notice also that we implicitly define three types of ‘‘diagonalizable’’ systems, depending on three kinds of transformation admitted. Definition 9. (a) A system is DEC (diagonalizable by strict equivalence over C) if there exist nonsingular U; V 2 C2n2n such that ^ UðlA BÞV ¼ lA^ B, ^ þ lD^ þ K. ^ ^ lÞ ¼ l2 M where lA^ B^ is the linearization of a (generally complex) diagonal system Lð 2n2n such that (b) A real system is DER if there exist nonsingular U; V 2 C ^ UðlA BÞV ¼ lA^ B, ^ þ lD^ þ K. ^ ^ lÞ ¼ l2 M where lA^ B^ is the linearization of a real diagonal system Lð (c) A system is DCR (diagonalizable by congruence) if there exists a nonsingular U 2 C2n2n such that ^ UðlA BÞU ¼ lA^ B, ^ þ lD^ þ K. ^ ^ lÞ ¼ l2 M where lA^ B^ is the linearization of a real diagonal system Lð
ARTICLE IN PRESS P. Lancaster, I. Zaballa / Mechanical Systems and Signal Processing 23 (2009) 1134–1144
1141
^ B^ and, in particular, because K^ Notice that, in case (c), if the system is Hermitian (or real and symmetric), then so are A; ^ are diagonal, so is B. ^ The following result now seems natural: and M Theorem 10. (a) A system LðlÞ with Jordan form J is DEC if and only if J 2 Jn;C . (b) A real system LðlÞ with Jordan form J is DER if and only if J 2 Jn;R . (c) An Hermitian system LðlÞ with Jordan form J is DCR if and only if J 2 Jn;R . Proof. For part (a): If LðlÞ is DEC then, in Definition 9(a), lA B and lA^ B^ have the same Jordan form, J, (it is preserved by strict equivalence), and since lA^ B^ is the linearization of a diagonal system, J 2 Jn;C . ^ lÞ. Thus, the Conversely, LðlÞ has Jordan form J 2 Jn;C implies that there is a strictly isospectral diagonal system Lð ^ ^ corresponding pencils lA B and lA B are isospectral. But then it follows from the Kronecker reduction of regular pencils ([14, Theorem 3.1], for example) that the pencils are strictly equivalent to the same canonical form, and hence to one another. (b) If LðlÞ is a real system which is DER then, in Definition 9(b), lA B and lA^ B^ have the same Jordan form J and, because lA^ B^ is generated by a real diagonal system, J 2 Jn;R , as required. The converse argument is as in (a) but over the real field; i.e. U and V can be taken with real entries [14, Theorem 3.2]. (c) If LðlÞ is an Hermitian system which is DCR then in Definition 9(c), lA B and lA^ B^ have the same Jordan form J and, because lA^ B^ is generated by a real diagonal system, J 2 Jn;R , as required. Conversely, let the Hermitian system LðlÞ have Jordan form J 2 Jn;R . We are to prove that there is a real diagonal system ^ lÞ such that the corresponding linearizations lA B and lA^ B^ are congruent. Lð Let the sign characteristic of LðlÞ be e (i.e. a fixed collection of þ1’s and 1’s associated with the partial multiplicities of the real eigenvalues.3See [8, Theorem 3.7] or [9, Theorem 12.5], for example). Then J 2 Jn;R implies that ^ lÞ. According to (13) the partial multiplicities of the eigenvalues of LðlÞ there are isospectral real diagonal systems Lð ^ (and then of LðlÞ) are either 2 or 1. By Proposition 10.12 in [9]4 we conclude that, for the real semisimple eigenvalues (i.e. those with only linear elementary divisors) the number of signs þ1 is equal to the number of ^ lÞ with distinct real zeros necessarily combine pairs of eigenvalues with signs 1. Furthermore, the diagonal terms of Lð opposite signs. ^ lÞ, multiplication by a diagonal of þ1’s and 1’s and exchanging factors corresponding to Now, given one such Lð semisimple real eigenvalues along the diagonal generates another isospectral diagonal system. Using this freedom, and knowing the signs attached to the real eigenvalues of LðlÞ, corresponding signs can be associated with the real ^ lÞ. In this way an Lð ^ lÞ is determined which is strictly isospectral with LðlÞ (in the terminology eigenvalues of Lð of [13]). ^ lÞ, respectively, and note that each one Now let lA B, lA^ B^ be the usual (Hermitian) linearizations of LðlÞ and Lð inherits both the spectrum and sign characteristic of the parent polynomial. Then it follows from Theorem 6.1 of [14] (attributed to Weierstrass) that the pencils lA B and lA^ B^ have the same canonical forms and are therefore congruent. Thus, LðlÞ is DCR. & We show next how the result of Theorem 10 can be applied to the system in Example 7. Example 8. Consider system LðlÞ in Example 7. The linearization of this system is 2
200
6 100 6 A¼6 4 1
3
2
100
1
0
100
0
0 2
0 0
27 7 7; 05
0
0
2000
6 1000 6 B¼6 4 0 0
3
1000
0
0
2000
0
0 0
1 0
07 7 7. 05 2
The aim is to obtain structure preserving transformations U and V for which UðlA BÞV ¼ lA^ B^ where 2 6 6 A^ ¼ 6 4
227:8131
0
1
0
3
2 6 6 B^ ¼ 6 4
0
22:1869
0
1
0
0
17 7 7; 05
0
1
0
0
3
2290:7119
0
0
0
0
654:8182
0
0
0
1
07 7 7 05
0
0
0
1
3 In Example 1 the eigenvalue l ¼ 1 has a partial multiplicity of order 4 with signature þ1. And in Example 3 L1 ðlÞ and L2 ðlÞ have two eigenvalues: l1 ¼ 1 and l2 ¼ 0. In addition, l1 has two partial multiplicities: 2 and 1. The signature of the eigenvalue l ¼ 1 with partial multiplicity 2 (i.e. the elementary divisor ðl 1Þ2 ) is þ1 for both systems. The signature for the elementary divisors ðl 1Þ is þ1 in system L1 ðlÞ and 1 in system L2 ðlÞ. Finally, the signature of the eigenvalue 0 is 1 in system L1 ðlÞ and þ1 in system L2 ðlÞ. 4
The result is proved here in the case M ¼ I, but extension to nonsingular Hermitian M is not difficult. Or see Proposition 4.2 of [7].
ARTICLE IN PRESS 1142
P. Lancaster, I. Zaballa / Mechanical Systems and Signal Processing 23 (2009) 1134–1144
^ lÞ. This is not an elementary matter but for this example and with the help of is the linearization of the diagonal system Lð MATLAB one can see that the following matrices work (notice that U and V are real): U= 1.0e+03* 0.0099 0.0007 0.0009 0.0003
0.0028 0.0002 0.0003 0.0005
2.1672 0.1878 0.2056 0.0057
0.6016 0.3224 0.0571 0.0112
20.4932 0.9530 216.0165 10.0342
0.0177 0.0296 0.4589 0.8731
0.0943 0.0044 0.9898 0.0450
0.0007 0.0013 0.0022 0
227.8131 0.0000 1.0000 0.0000
0.0000 22.1869 0.0000 1.0000
1.0000 0.0000 0.0000 0.0000
0.0000 1.0000 0.0000 0.0000
1.0e+03 * 2.2907 0.0000 0.0000 0.0000
0.0000 0.6548 0.0000 0.0000
0.0000 0.0000 0.0010 0.0000
0.0000 0.0000 0.0000 0.0010
V=
bU*A*V ans =
bU*B*V ans =
6. Conclusions A complete characterization has been given of n n quadratic systems LðlÞ which can be diagonalized by applying structure preserving transformations (congruence or strict equivalence) to the 2n 2n linearization lA B of the system (see (1)). This raises the familiar question of diagonalizing LðlÞ itself by strict equivalence or congruence. This theory has been reviewed and improved. This paper is part of a more general research program with the goal of parametrization of structure preserving transformations which, in turn, can be used in the design of filters (see [6]). These filters, which transform LðlÞ to an isospectral diagonal form, are to be physically implemented. A natural extension of the theory would be to the diagonalization of n n systems of higher order, r42. To accomplish this in the n-dimensional space of the system itself would be unrealistic for most applications. However, diagonalization of linearizations would be feasible and admit greater flexibility (in terms of Jordan structures) as r increases.
Acknowledgements This project was partially funded by EPSRC (UK) Grant EP/E046290. The authors are duly grateful to the EPSRC and to their partners in this project, S.D. Garvey, U. Prells and A.A. Popov, of the University of Nottingham for constant support and useful advice. The authors are also grateful to a reviewer whose comments led to a significant improvement in exposition. Appendix A. Canonical forms for systems that can be decoupled Given Theorem 6 and the complete descriptions of the admissible Jordan forms Jn;C and Jn;R for systems that can be decoupled, we can obtain the canonical forms for the corresponding linearizations lA B of (1). In particular, since we assume that M is nonsingular, attention can be confined to the tractable cases in which A is nonsingular. The classes of transformations admitted are: (1) Strict equivalence over C. (2) Strict equivalence over R. (3) Congruence over C. (4) Congruence over R. The review of [14] will assist in this task. A.1. Complex systems—no symmetry: strict equivalence over C [14, Theorem 3.1] There is a canonical pencil of the form s M j¼1
ðlI ‘j J ‘j ðlj ÞÞ.
ARTICLE IN PRESS P. Lancaster, I. Zaballa / Mechanical Systems and Signal Processing 23 (2009) 1134–1144
1143
Thus, this form is determined entirely by the Jordan form. The only constraints on the parameters are that ‘j p2 for each j P and sj¼1 ‘j ¼ 2n. A.2. Hermitian systems: strict equivalence over C [14, Theorem 5.1] The sizes of Jordan blocks for the real eigenvalues are not greater than two and for nonreal eigenvalues they have size one. Suppose that there are (a) real eigenvalues a1 ; . . . ; ar with a partial multiplicity equal to one, (b) real eigenvalues arþ1 ; . . . ; as with a partial multiplicity equal to two, and (c) nonreal conjugate pairs msþ1 iosþ1 ; . . . ; mt iot with partial multiplicity one. Then there is a canonical pencil of the form " # " r s t 0 M M M 1 l aj ðl aj Þ l ðmj ioj Þ l aj 0 j¼1
j¼rþ1
l ðmj þ ioj Þ 0
j¼sþ1
# .
A.3. Hermitian systems: congruence over C [14, Theorem 6.1] With the same conventions as Section A.2, there is a canonical pencil of the form " # " # r s t 0 l ðmj þ ioj Þ M M M 1 l aj Zj ðl aj Þ Zj . l ðmj ioj Þ 0 l aj 0 j¼1
j¼rþ1
j¼sþ1
The numbers Z1 ; . . . ; Zs take the values þ1 or 1. They form the sign characteristic of the system. The canonical forms of Cases 2 and 3 differ only in this respect. A.4. Real systems—no symmetry: strict equivalence over R [14, Theorem 3.2] Description of canonical forms over R requires the introduction of two more standard forms. (They will account for the presence of nonreal eigenvalues.) For real numbers, m and oa0 define real Jordan blocks by 2 3 m o 1 0 " # 6 7 m o 0 17 6 o m 7. ; J 2 ðm ioÞ ¼ 6 J 1 ðm ioÞ ¼ 6 0 o m m o7 0 4 5 0 0 o m (The subscripts 1 and 2 refer to the partial multiplicities of eigenvalues of the form m io). Suppose that there are (a) (b) (c) (d)
real eigenvalues a1 ; . . . ; ar with a partial multiplicity equal to one; real eigenvalues arþ1 ; . . . ; as with a partial multiplicity equal to two; nonreal conjugate pairs msþ1 iosþ1 ; . . . ; mt iot with partial multiplicity one; nonreal conjugate pairs mtþ1 iotþ1 ; . . . ; mu iou with partial multiplicity two.
Then there is a canonical pencil of the form " # r s t u M M M M l aj 1 ðl aj Þ ðlI 2 J 1 ðmj ioj ÞÞ ðlI 4 J 2 ðmj ioj ÞÞ. l aj 0 j¼1
j¼rþ1
j¼sþ1
j¼tþ1
A.5. Real-symmetric systems: strict equivalence over R [14, Theorem 9.1] For real numbers m and oa0 define 2
" K 1 ðm ioÞ ¼
#
o m ; m o
0 6 61 K 2 ðm ioÞ ¼ 6 6o 4
1 0
m m o
3
o m m o 7 7 0 0
7. 0 7 5 0
ARTICLE IN PRESS 1144
P. Lancaster, I. Zaballa / Mechanical Systems and Signal Processing 23 (2009) 1134–1144
Then with the same conventions as Case 4, there is a real-symmetric canonical pencil of the form " # r s t u M M M M 1 l aj ðl a j Þ ðlI 2 K 1 ðmj ioj ÞÞ ðlI 4 K 2 ðmj ioj ÞÞ. l aj 0 j¼1
j¼rþ1
j¼sþ1
j¼tþ1
A.6. Real-symmetric systems: congruence over R [14, Theorem 9.2] With the same conventions as Section A.4, there is a real-symmetric canonical pencil of the form " # r s t u M M M M 1 l aj Zj ðl aj Þ Zj ðlI 2 K 1 ðmj ioj ÞÞ ðlI 4 K 2 ðmj ioj ÞÞ. l aj 0 j¼1
j¼rþ1
j¼sþ1
j¼tþ1
The numbers Z1 ; . . . ; Zs take the values þ1 or 1. They form the sign characteristic of the system. The canonical forms of Cases 5 and 6 differ only in this respect. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [13] [14] [15] [16] [17]
T.K. Caughey, M.E.J. O’Kelly, Classical normal modes in damped linear dynamic systems, ASME J. Appl. Mech. 32 (1965) 583–588. M. Chu, N. Del Buono, Total decoupling of general quadratic pencils, Part I: theory, J. Sound Vibr. 309 (2008) 96–111. M. Chu, N. Del Buono, Total decoupling of general quadratic pencils, Part II: structure preserving isospectral flows, J. Sound Vibr. 309 (2008) 112–128. S.G. Garvey, M.I. Friswell, U. Prells, Co-ordinate transformations for second order systems, Part 1: general transformations, J. Sound Vibr. 258 (2002) 885–909. S.G. Garvey, M.I. Friswell, U. Prells, Z. Chen, General isospectral flows for linear dynamic systems, Linear Algebra Appl. 385 (2004) 335–368. S.G. Garvey, Achieving stable diagonalising filters for second order systems, in: Proceedings of the XXVI IMA Conference, February, 2008. I. Gohberg, P. Lancaster, L. Rodman, Spectral analysis of selfadjoint matrix polynomials, Research Paper 419, Department of Mathematics and Statistics, University of Calgary, 1979. I. Gohberg, P. Lancaster, L. Rodman, Spectral analysis of selfadjoint matrix polynomials, Ann. Math. 112 (1980) 33–71. I. Gohberg, P. Lancaster, L. Rodman, Matrix Polynomials, Academic Press, New York, 1982. P. Lancaster, Some questions in the classical theory of vibrating systems, Bull. Poly. Inst. Jassy 17 (1971) 111–112. P. Lancaster, Inverse spectral problems for semisimple damped vibrating systems, SIAM J. Matrix Anal. Appl. 20 (2007) 279–301. P. Lancaster, U. Prells, Isospectral families of high-order systems, Z. Angew. Math. Mech. 87 (2007) 219–234. P. Lancaster, L. Rodman, Canonical forms for Hermitian matrix pairs under strict equivalence and congruence, SIAM Rev. 47 (2005) 407–443. M. Liu, J.M. Wilson, Criterion for decoupling dynamic equations of motion of linear gyroscopic systems, AIAA J. 30 (1992) 2989–2991. F. Ma, T.K. Caughey, Analysis of linear nonconservative vibrations, ASME J. Appl. Mech. 62 (1995) 685–691. K. Meerbergen, Fast frequency response computation for Rayleigh damping, Int. J. Numer. Methods Eng. 73 (1) (2008) 96–106.
Contents lists available at ScienceDirect
Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/jnlabr/ymssp
Diagonalizable quadratic eigenvalue problems Peter Lancaster a,1, Ion Zaballa b,,2 a b
Department of Mathematics and Statistics, University of Calgary, Calgary, AB, Canada T2N 1N4 ´tica Aplicada y EIO, Euskal Herriko Univertsitatea, Apdo. Correos 644, 48080 Bilbao, Spain Departamento de Matema
a r t i c l e i n f o
abstract
Article history: Received 30 May 2008 Received in revised form 3 November 2008 Accepted 6 November 2008 Available online 30 November 2008
A system is defined to be an n n matrix function LðlÞ ¼ l M þ lD þ K where M; D; K 2 Cnn and M is nonsingular. First, a careful review is made of the possibility of direct decoupling to a diagonal (real or complex) system by applying congruence or strict equivalence transformations to LðlÞ. However, the main contribution is a complete description of the much wider class of systems which can be decoupled by applying congruence or strict equivalence transformations to a linearization of a system while preserving the structure of LðlÞ. The theory is liberally illustrated with examples. & 2008 Published by Elsevier Ltd.
2
Keywords: Damped systems Decoupling Diagonalization Linearization
1. Introduction 2
A system is defined to be a matrix function Ll ¼ l M þ lD þ K where M; D; K 2 Cnn and M is nonsingular. If M; D; K 2 Rnn then we have a real system. Similarly, if M; D; K are all Hermitian, or all real and symmetric then the system is said to be Hermitian or real symmetric, respectively. The system is said to be diagonal or decoupled if M; D and K are diagonal matrices. In general M, D and K arise from the modelling of three independent physical phenomena. Two systems will be called isospectral if they share the same Jordan form; i.e. the same eigenvalues and the same partial multiplicities. Thus, a system is diagonalizable or, equivalently, it can be decoupled if it admits an isospectral diagonal system. There are some diagonalizable systems for which the diagonal form can be achieved by either a congruence transformation, LðlÞ ! U LðlÞU for some nonsingular matrix U, or a strict equivalence transformation, LðlÞ ! ULðlÞV for nonsingular matrices U and V . These are well understood but, even so, some improvement in the existing theory are made here in Section 2. Those relatively simple cases require that, in effect, one of the coefficients M, D, K is expressed in terms of the other two and their natural independence is lost. A more general diagonalization process is the main subject of this paper. The strategy is to apply certain congruence or strict equivalence transformations to the familiar isospectral ‘‘linearization’’ lA B of LðlÞ, where A¼
D
M
M
0
;
B¼
K
0
0
M
,
(1)
Corresponding author.
E-mail addresses: [email protected] (P. Lancaster), [email protected] (I. Zaballa). Partially supported by the Natural Sciences and Engineering Research Council of Canada. 2 Partially supported by the Direccio´n General de Investigacio´n Cientı´fica y Te´cnica, Proyecto de Investigacio´n MTM2007-67812-C02-01 and Gobierno Vasco GIC07/154-IT-327-07. 1
0888-3270/$ - see front matter & 2008 Published by Elsevier Ltd. doi:10.1016/j.ymssp.2008.11.007
ARTICLE IN PRESS P. Lancaster, I. Zaballa / Mechanical Systems and Signal Processing 23 (2009) 1134–1144
1135
and observe that, if M, D and K are hermitian or real symmetric, then so are A and B. This is achieved by confining attention to those transformations which preserve the block structure of A and B (the so-called structure preserving transformations). Algorithms of this kind have been studied by Garvey et al. [4,5] and by Chu and Del Buono [2,3], and it is claimed that ‘‘almost all’’ systems LðlÞ can be diagonalized in this way. Our main result is Theorem 10, in which this claim is verified and made precise in three cases: general complex or real systems and structure preserving strict equivalence, and hermitian systems and structure preserving congruence. In other words, if any system of these types is diagonalizable, then it can be diagonalized by applying the corresponding structure preserving transformation to the linearization. The key question now is what systems are diagonalizable. The answer is given in Section 3 providing a complete description of the admissible Jordan forms in such case. In particular, diagonal Jordan forms (semisimple systems) are included among them, confirming the earlier claim that ‘‘almost all’’ systems are diagonalizable. Several informative examples are included for illustration. Real application and physical implementation of these concepts are expected in the near future through the design of filters (see [6]). Our analysis depends on detailed knowledge of the theory of reduction of matrix pairs by congruence or strict equivalence transformations. For this, we rely on the recent comprehensive survey of Lancaster and Rodman [14]. However, it may be helpful to introduce some relevant ideas here. When reducing a Hermitian or real-symmetric pencil lA B by congruence without assuming that the leading coefficient is positive definite, knowledge of the notion of the sign characteristic of each real eigenvalue is required. (It appears in Sections A.3 and A.6 of the Appendix, for example, as the set of numbers Zj .) Each real eigenvalue has one or more ‘‘partial multiplicities’’ (the sizes of Jordan blocks) and a þ1 or 1 is associated with each of them. The eigenvalue has positive type if all the associated numbers are þ1, and similarly for eigenvalues of negative type. An eigenvalue has definite type if it is either of positive or negative type and, otherwise, it has mixed type. The geometric multiplicity of an eigenvalue is the number of its partial multiplicities and its algebraic multiplicity is the sum of the partial multiplicities. For completeness, and for the reader’s convenience, an Appendix is provided giving a summary of canonical forms for linearizations under either congruence or strict equivalence transformations. 2. Diagonalization without linearization There are many engineering applications in which the system is Hermitian and M is positive definite (M40) and, in the absence of a good mathematical model of the damping phenomenon, D is supposed to be a linear combination of M and K (the hypothesis of ‘‘proportional damping’’). Some straightforward generalizations arise frequently in the engineering and computational literature and are variously known as ‘‘modal’’ or ‘‘Rayleigh’’ damping. See [17], for example, for an interesting recent study. Several variations on the same theme can arise according as strict equivalence or congruence transformations are used, the coefficients are real or complex, and with or without symmetry. But in every case, if three matrices are to be diagonalized simultaneously then the Caughey–O’Kelly commutativity condition DM 1 K ¼ KM 1 D, or something equivalent, is required (see [1,15,16]). 2.1. Hermitian systems: reduction by congruence When a system has Hermitian coefficients and the condition M40 is relaxed (as below), then real eigenvalues can arise and, as described above, they have either positive type, negative type, or mixed type. (See [14] or Appendix B of [11], for example.) Here, we first admit semisimple (i.e. nondefective) real eigenvalues with no restriction on the type. Lemma 1. Let M; K 2 Cnn with det Ma0, M ¼ M, K ¼ K. Assume that lM þ K is semisimple with all eigenvalues real and define
L ¼ diag½l1 I 1 ; l2 I 2 ; . . . ; ls I s ;
S ¼ diag½I 1 ; I 2 ; . . . ; I s ,
(2)
where the size of the identity matrix I j is a partial multiplicity of eigenvalue lj for each j, and the sign of each term in S is determined by the corresponding þ1 or 1 in the sign characteristic. Then there exists a family of nonsingular matrices V 2 Cnn such that V MV ¼ S;
V KV ¼ S L.
(3)
If V is one such matrix, then so is any matrix VA where A ¼ diag½A1 ; A2 ; . . . ; As and each Aj is unitary with the size of I j . Proof. This is a classical result. It is a special case of Theorem 6.1 of [14]. (Note that there may be repetitions among the lj .) & A result closely analogous to Lemma 1 holds in the case that M; K 2 Rnn . It is only necessary to use congruence over R and to replace the unitary matrices Aj by real orthogonal matrices. (This is a special case of Theorem 9.2 of [14].) Note carefully that the further condition M40 would ensure that all eigenvalues are real and of positive type,
ARTICLE IN PRESS 1136
P. Lancaster, I. Zaballa / Mechanical Systems and Signal Processing 23 (2009) 1134–1144
and then S ¼ I. This is the case in the original paper of Caughey and O’Kelly of 1965 [1]. Their theorem can be generalized as follows: Theorem 2. Let the hypotheses of Lemma 1 hold, assume also that all eigenvalues of lM þ K have definite type, and that D ¼ D. Then there exists a nonsingular U 2 Cnn such that U MU, U DU, and U KU are diagonal if and only if DM 1 K ¼ KM 1 D. If, in addition, M; D; K are real and symmetric, then there is a corresponding U 2 Rnn . Proof. We use the notations introduced in Lemma 1. From (3) we obtain M 1 K ¼ ðVSV ÞðV S LV 1 Þ ¼ VS 2 LV 1 ¼ V LV 1 ,
(4)
KM 1 ¼ ðV S LV 1 ÞðVSV Þ ¼ V S LSV ¼ V LU .
(5)
If also DM
1
K ¼ KM
1
D then DV LV
1
¼V
LV D and so
ðV DV ÞL ¼ LðV DV Þ. The assumption that all eigenvalues have definite type means that the eigenvalues l1 ; . . . ; ls of (2) are distinct. And so it follows that V DV is block-diagonal (as in (2)). But, as the blocks of V DV are Hermitian, the unitary blocks of matrix A of Lemma 1 can be chosen to further reduce V DV to diagonal form. Conversely, if M 0 :¼U MU; D0 :¼U DU; K 0 ¼ U KU are diagonal, it is easily verified that DM 1 K ¼ KM 1 D. The case of real-symmetric matrices M; D; K is very similar and depends on the analogue of Lemma 1 mentioned above. & 2.2. No symmetry: reduction by strict equivalence For systems which are not symmetric (or Hermitian) it is natural to replace the congruence transformations of LðlÞ used above by strict equivalence transformations. This possibility was investigated by Ma and Caughey [16] and it is interesting that, in this case too, the commutativity condition of Caughey–O’Kelly continues to play a restrictive role. However, Theorem 3 of [16] is false as it stands—a stronger hypothesis is required. A counter-example for that theorem is 0 1 1 0 2 1 0 LðlÞ ¼ l þl þ . 0 1 0 0 0 1 One possibility is to replace the condition that lM þ K is semisimple by requiring distinct eigenvalues for this pencil (which is obviously not satisfied by the counter-example). Lemma 3. Let M; K 2 Cnn with det Ma0, assume that lM þ K is semisimple, and write a diagonal matrix of the eigenvalues of lM þ K in the form
L ¼ diag½l1 I 1 ; l2 I 2 ; . . . ; ls I s , where li alj when iaj. Then there is a family of nonsingular matrices U; V 2 Cnn such that UMV ¼ I
and
UKV ¼ L.
(6) 1
If A ¼ diag½A1 ; . . . ; As is nonsingular and Aj has the size of I j , then U; V can be replaced by A
U; VA; respectively.
Proof. This is the classical result known as reduction to Kronecker form. It is a special case of Theorem 3.1 of [14].
&
Once again, there is a real analogue of this result. The reader will be able to formulate this analogue (for real strict equivalence transformations of real systems) using Theorem 3.2 of [14], for example. Theorem 4. Let M; D; K 2 Cnn with det Ma0 and assume that lM þ K has n distinct eigenvalues. Then there exist nonsingular U; V 2 Cnn such that UMV ¼ I, and UDV , UKV are diagonal if and only if DM 1 K ¼ KM 1 D. Proof. First use Lemma 3 to obtain nonsingular U; V 2 Cnn such that UMV ¼ I and UKV ¼ L, a diagonal matrix. Then M 1 K ¼ ðVUÞðU 1 LV 1 Þ ¼ V LV 1 ,
(7)
KM 1 ¼ ðU 1 LV 1 ÞðVUÞ ¼ U 1 LU. If also DM
1
K ¼ KM
1
D then DðV LV
1
(8) Þ ¼ ðU
1
LUÞD and hence (9)
ðUDV ÞL ¼ LðUDV Þ. Since L is diagonal with distinct diagonal entries, this implies that UDV is also diagonal, as required. Conversely, if M 0 :¼UMV ; D0 :¼UDV ; K 0 ¼ UKV are diagonal, it is easily verified that DM 1 K ¼ KM 1 D.
&
ARTICLE IN PRESS P. Lancaster, I. Zaballa / Mechanical Systems and Signal Processing 23 (2009) 1134–1144
1137
If the assumption that lM þ K has distinct eigenvalues is relaxed and we assume only that lM þ K is semisimple (as in [16]), then the proof of the last theorem can be followed as far as Eq. (9). One may then be able to use the freedom of choice of matrix A of Lemma 3 to reduce the blocks of UDV by similarity (the strategy proposed in [16]). This will be possible provided that UDV (and hence each diagonal block of UDV ) is semisimple. 3. Jordan forms for diagonal systems As shown in the previous section, most systems cannot be diagonalized by strict equivalence or congruence; a very strong commutativity condition must be satisfied. However, as we see now, most systems admit diagonal isospectral systems; i.e. diagonal systems with the same Jordan form as the original one. As in the Introduction, these systems will be called diagonalizable and we aim to completely describe them through their Jordan forms. Once a system is known to be diagonalizable, the natural question is how to reduce it to diagonal form. It is shown in Section 5 that this can always be achieved by means of structure preserving transformations applied to the linearizations lA B given in (1). In order to describe the Jordan forms of diagonalizable systems, we consider the primitive case of systems where the coefficients M; D; K are already diagonal. They can be seen as the ‘‘target’’ systems to which more general systems are to be reduced (whenever possible). The Jordan form for a system LðlÞ is the classical canonical form for A1 B under similarity transformations over C (used here unless specified otherwise), or over R. Definition 5. Let Jn;C and Jn;R be the classes of 2n 2n canonical Jordan matrices for n n diagonal systems, and n n real diagonal systems, respectively (so that Jn;R Jn;C C2n2n ). A Jordan block with eigenvalue lj and size s is denoted by 2 3 lj 1 0 0 60 l 1 0 7 j 6 7 6 7 6 .. .. 7 . 7 J s ðlj Þ:¼6 6 . 7. 6 . 7 6 .. 0 17 4 5 0 0 0 lj L It will be convenient to denote a direct (diagonal) sum of scalars or matrices, x1 ; . . . ; xk by kj¼1 xj. It is clear that, given any set of 2n complex numbers, they can be sorted into n pairs and n scalar quadratics are determined, each having one pair of the given numbers as its zeros. Diagonal Jordan matrices in the class Jn;C can be formed in this way. However, a similar construction for Jn;R requires that the 2n numbers consist of, say, r pairs of real numbers (0prpn) and n r pairs of (nonreal) conjugate complex numbers. As long as all 2n numbers are distinct the corresponding Jordan forms will be diagonal, but the situation becomes more complicated if repetitions among the eigenvalues are permitted. Assume that there exist distinct eigenvalues l1 ; . . . ; lt 2 C; 1ptp2n, and let li have partial multiplicities ni1 X Xni;mg;i 40 for each i (forming the ‘‘Segre characteristic’’). Then eigenvalue li has geometric multiplicity mg;i pn Pmg;i and algebraic multiplicity ma;i ¼ j¼1 nij p2n. Also, mg;i t X X
nij ¼ 2n.
(10)
i¼1 j¼1
L Q 2 Write LðlÞ ¼ ni¼1 ½mi l þ di l þ ki ; where ni¼1 mi a0. Then each diagonal entry has a linearization " # 0 1 lI 2 ; i ¼ 1; 2; . . . ; t, ki =mi di =mi and LðlÞ has the tridiagonal linearization lI A where " # n M 0 1 A¼ . ki =mi di =mi
(11)
(12)
i¼1
Furthermore, the elementary divisors of lI A are just the disjoint union of those of (11). Hence, 1pnij p2
for 1pipt; 1pjpmg;i .
For each distinct eigenvalue li , i.e. for i ¼ 1; 2; . . . ; t, we define the integers si X0 by writing ( 2 for j ¼ 1; 2; . . . ; si ; nij ¼ 1 for j ¼ si þ 1; . . . ; mg;i :
(13)
(14)
Each partial multiplicity nij ¼ 2 is necessarily associated with just one block of (12) and so, if p ¼ s1 þ s2 þ þ st (the number of quadratic elementary divisors) then the remaining n p diagonal blocks of A cannot have repeated eigenvalues
ARTICLE IN PRESS 1138
P. Lancaster, I. Zaballa / Mechanical Systems and Signal Processing 23 (2009) 1134–1144
because they are nonderogatory matrices. Thus, in (14) we have
mg;i si pn p;
i ¼ 1; 2; . . . ; t.
(15)
We now claim: i¼t;j¼m
Theorem 6. A Jordan matrix J with Segre characteristic fnij gi¼1;j¼1g;i (defined as above) is in Jn;C if and only if conditions (10), (13) and (15) hold where, for i ¼ 1; 2; . . . ; t the integers si X0 appearing in (15) are defined by (14). Proof. The necessity of the three conditions has been established. Suppose, conversely, that a Jordan matrix satisfies (10), (13) and (15). As above, the distinct eigenvalues of J are l1 ; . . . ; lt with geometric multiplicities mg;1 ; . . . ; mg;t , respectively, and s1 ; . . . ; st are the respective numbers of partial multiplicities equal to 2. Furthermore, it may be assumed that
mg;1 s1 Xmg;2 s2 X Xmg;t st . There are second degree elementary divisors for i ¼ 1; 2; . . . ; t and j ¼ 1; 2; . . . ; si . Let Aij be their corresponding 2 2 companion matrices. The remaining elementary divisors are linear and their eigenvalues can be sorted into distinct pairs, say ðlki ; lkj Þ with lki alkj . Since the number of elementary divisors of degree two is p:¼s1 þ s2 þ þ st , the number of linear elementary P divisors is 2ðn pÞ ¼ ti¼1 ðmg;i si Þ. Conditions (15) and (10) ensure that the corresponding eigenvalues can be sorted into n p pairs ðlki ; lkj Þ with lki alkj . In other words, the eigenvalues with linear elementary divisors can be organized in two ordered lists, each with n p eigenvalues, say and
ðapþ1 ; apþ2 ; . . . ; an Þ
ðbpþ1 ; bpþ2 ; . . . ; bn Þ
with ai abi for i ¼ p þ 1; . . . ; n. For each such i let Bi be the companion matrix of ðl ai Þðl bi Þ and observe that each Bi is diagonalizable. Now the matrix A ¼ diagðA11 ; . . . ; A1s1 ; . . . ; At1 ; . . . ; Atst ; Bpþ1 ; . . . ; Bn Þ has J for its Jordan form and, also, lI 2n A is the linearization of a (monic) diagonal matrix polynomial of degree 2, i.e. J 2 Jn;C . & It follows immediately from (13) that a necessary, but not sufficient condition that J 2 Jn;C is
ma;i p2mg;i for i ¼ 1; 2; . . . ; t.
(16)
Each one of the following examples shows a particular feature related to the possibility of diagonalizing the given system. Example 1. Let 2 1 LðlÞ ¼ l 0
0 2 þl 1 1
1 1 þ 2 1
1 . 2
It is easily seen that det LðlÞ ¼ ðl þ 1Þ4 so there is just one eigenvalue, 1, with ma ¼ 4. Also, Lð1Þ ¼ ½00 10, so mg ¼ 1. It follows that J consists of just one Jordan block of size four. Conditions (13) and (16) above are not satisfied, so JeJn;C and the system is not diagonalizable. The commutativity condition DM 1 K ¼ KM 1 D may or may not hold (in this case it does not hold but there are nondiagonalizable systems for which this condition is satisfied, see Example 2), so Theorem 2 does not apply. Example 2. Let 2
3
2
1
1
1
1
26 LðlÞ ¼ l 4 1 1
0 1
6 7 1 5 þ l4 1 3=2 0 2
1 0 2
3=2
3
2
0
6 2 7 5þ4 0 0 1=2
4
0 0 1
1=2
3
1 7 5. 0
In this case det LðlÞ ¼ l ðl 1Þ . The eigenvalue þ1 has ma ¼ 4 and the rank of Lð1Þ is 0. Thus mg ¼ 3 and condition (16) is 2 fulfilled. For this system, however, JeJn;C because the elementary divisors of LðlÞ are ðl 1Þ2 , ðl 1Þ, ðl 1Þ and l and condition (15) is not satisfied. Although DM 1 K ¼ KM 1 D in this case, LðlÞ cannot be decoupled by strict equivalence; otherwise, it would have a Jordan form satisfying condition (15). Also, Theorems 2 and 4 do not apply because M 1 K has the defective eigenvalue zero with algebraic multiplicity ma ¼ 2 and geometric multiplicity mg ¼ 1. Example 3. The following two systems: 73 36 32 2 41 12 L1 ðlÞ ¼ l þl þ 12 34 36 52 24
24 18
ARTICLE IN PRESS P. Lancaster, I. Zaballa / Mechanical Systems and Signal Processing 23 (2009) 1134–1144
1139
and 2
L2 ðlÞ ¼ l
0 1
1 þl 3 3
1
3 7
þ
1
2
2
4
have the same Jordan structure. In fact, det L1 ðlÞ and det L2 ðlÞ are scalar multiple of lðl 1Þ3 . So, for the eigenvalue þ1, ma ¼ 3 and since L1 ð1Þ ¼ L2 ð1Þ ¼ 0, this eigenvalue has geometric multiplicity mg ¼ 2. Actually, for both systems the elementary divisors are: ðl 1Þ2 , ðl 1Þ and l. Thus, all conditions of Theorem 6 are satisfied and L1 ðlÞ, L2 ðlÞ and the diagonal system 1 0 1 0 0 0 ^ lÞ ¼ l2 Lð þl þ 0 1 0 2 0 1 are isospectral. However, there is an important difference between L1 ðlÞ and L2 ðlÞ: the mass matrix in L1 ðlÞ is positive definite and in ^ lÞ are ‘‘strictly’’ isospectral, in the sense of [13] (the sign characteristics are the L2 ðlÞ it is not. It turns out that L1 ðlÞ and Lð ^ same), and L2 ðlÞ and LðlÞ are not strictly isospectral. ^ lÞ are In addition, for all three systems DM 1 K ¼ KM 1 D. In particular, Theorem 2 applies for L1 ðlÞ; i.e. L1 ðlÞ and Lð congruent. Also, Theorem 4 applies for L2 ðlÞ. That is to say, L2 ðlÞ can be reduced to monic diagonal form by real strict equivalence. Example 4. Let 5 2 LðlÞ ¼ l 2
2
1
319 þl 126 2
2666 1053
þ
0
319
0
126
.
2
For this system det LðlÞ ¼ l ðl 1Þ. For the eigenvalue 0, ma ¼ 2 and mg ¼ 1 and for the eigenvalues þ1 and 1, ma ¼ mg ¼ 1. Thus all conditions of Theorem 6 hold and an isospectral diagonal system is, for example, 2 0 0 0 ^ lÞ ¼ l2 Lð þ . 0 1 0 1 However, although lM þ K has distinct eigenvalues, DM 1 KaKM 1 D and so,by Theorem 4, LðlÞ cannot be decoupled by strict equivalence. 4. Jordan forms for real diagonal systems Now consider Jordan canonical forms for real diagonal systems. i¼t;j¼m
Theorem 7. A Jordan matrix J with Segre characteristic fnij gi¼1;j¼1g;i (defined as above) is in Jn;R if and only if there is an n0 , 0pn0 pn, such that J ¼ diagðJ n0 ; J nn0 Þ for Jordan matrices J n0 ; J nno with sðJ n0 Þ R and sðJ nn0 Þ \ R ¼ ; and: (a) conditions (10), (13) and (15) (with n replaced by n0 ) hold for J n0 and (b) sðJ nn0 Þ consists of conjugate pairs of nonreal semisimple eigenvalues lj ; lj . Note that, if the system (and hence J) has no real eigenvalues then n0 ¼ 0 and J n0 simply does not appear, and if the system (and hence J) has no nonreal eigenvalues then n0 ¼ n and J nn0 does not appear. To illustrate, in the case of Example 4, n0 ¼ n and so J nn0 does not appear and 2 3 0 1 0 0 60 0 0 0 7 6 7 J ¼ J n0 ¼ 6 7. 40 0 1 0 5 0
0
0
1
Proof. If LðlÞ is a real diagonal system and l0 is an eigenvalue with l0 al0 , then unit co-ordinate vectors ej can be chosen as corresponding eigenvectors, and the number of such independent vectors is just the algebraic multiplicity of l0 . Thus, l0 is semisimple, and similarly for all nonreal eigenvalues. After permutation, the set of all nonreal eigenvalues determines the matrix J nn0 satisfying condition (b). For the complementary part of J all eigenvalues are real and the argument used in the proof of Theorem 6 can be applied to obtain condition (a). (The argument over the real field is the same as that over the complex numbers used in the earlier proof.) The sufficiency of the prescribed conditions is clear. &
ARTICLE IN PRESS 1140
P. Lancaster, I. Zaballa / Mechanical Systems and Signal Processing 23 (2009) 1134–1144
Systems with all eigenvalues semisimple clearly satisfy the conditions of Theorem 6, if complex, or Theorem 7, if real. So, we have Corollary 8. If all eigenvalues of LðlÞ are semisimple and J is its Jordan form then J 2 Jn;C if LðlÞ is a complex system and J 2 Jn;R if LðlÞ is real. Example 5 (See Lancaster [10, Example 13.1]). This example shows that, in contrast to condition (b) above, a real-symmetric system may have nonreal eigenvalues with nonlinear elementary divisors. Thus, " pffiffiffi # 1 3=2 1 0 2 1 0 l þ pffiffiffi lþ LðlÞ:¼ (17) 3=2 2 0 1 0 4 pffiffiffiffiffiffi has eigenvalues 14ð3 i 23Þ with elementary divisors of degree two. Example 6. The following system, however, satisfies the conditions of Theorem 7: " # " # 3=2 1=2 2 11=2 9=2 3 5 l þ lþ . LðlÞ ¼ 1=2 3=2 9=2 11=2 5 3 In fact, its eigenvalues are: l1 ¼ 1, l2 (i.e. ma ¼ 2 and mg ¼ 1), its Jordan form is 1 0 2 0 1 ^ lÞ ¼ l2 Lð þl þ 0 1 0 4 0
¼ 2 þ i and l3 ¼ 2 i. Since the eigenvalue 1 has partial multiplicity (2) in Jn;R . A diagonal isospectral system is 0 . 5
Furthermore, it is easily verified that DM 1 K ¼ KM 1 D. Since M is positive definite, Theorem 2 applies and LðlÞ can be decoupled by congruence (applied to LðlÞ itself) to obtain a real diagonal system. Example 7. The final example corresponds to a simple two-degree-of-freedom mass–spring system: 200 100 2000 1000 2 1 0 LðlÞ ¼ l þl þ . 0 2 100 100 1000 2000 This is not a proportionally damped system and Theorems 2 and 4 show that LðlÞ cannot be reduced to diagonal form by either congruence or strict equivalence. However, the eigenvalues of LðlÞ are all distinct: 217:2699, 10:5432, 11:0935 þ 23:0598i, 11:0935 23:0598i. By Corollary 8, the Jordan form of LðlÞ is in J n;R . A diagonal system with the same Jordan form is 227:8131 0 2290:7119 0 1 0 ^ lÞ ¼ l2 Lð þ þl . 0 22:1869 0 654:8182 0 1
5. Linearization and diagonalizable systems Given a system LðlÞ, we now consider the generation of an isospectral diagonal system (when one exists) by the application of strict equivalence, or congruence transformations to the linearization lA B (see (1)). The following specific classes of transformations will be considered. Notice that they are all ‘‘structure preserving’’ transformations in the sense that the block structures of (1) are preserved and, in each case, the transformations are defined over the complex numbers. Notice also that we implicitly define three types of ‘‘diagonalizable’’ systems, depending on three kinds of transformation admitted. Definition 9. (a) A system is DEC (diagonalizable by strict equivalence over C) if there exist nonsingular U; V 2 C2n2n such that ^ UðlA BÞV ¼ lA^ B, ^ þ lD^ þ K. ^ ^ lÞ ¼ l2 M where lA^ B^ is the linearization of a (generally complex) diagonal system Lð 2n2n such that (b) A real system is DER if there exist nonsingular U; V 2 C ^ UðlA BÞV ¼ lA^ B, ^ þ lD^ þ K. ^ ^ lÞ ¼ l2 M where lA^ B^ is the linearization of a real diagonal system Lð (c) A system is DCR (diagonalizable by congruence) if there exists a nonsingular U 2 C2n2n such that ^ UðlA BÞU ¼ lA^ B, ^ þ lD^ þ K. ^ ^ lÞ ¼ l2 M where lA^ B^ is the linearization of a real diagonal system Lð
ARTICLE IN PRESS P. Lancaster, I. Zaballa / Mechanical Systems and Signal Processing 23 (2009) 1134–1144
1141
^ B^ and, in particular, because K^ Notice that, in case (c), if the system is Hermitian (or real and symmetric), then so are A; ^ are diagonal, so is B. ^ The following result now seems natural: and M Theorem 10. (a) A system LðlÞ with Jordan form J is DEC if and only if J 2 Jn;C . (b) A real system LðlÞ with Jordan form J is DER if and only if J 2 Jn;R . (c) An Hermitian system LðlÞ with Jordan form J is DCR if and only if J 2 Jn;R . Proof. For part (a): If LðlÞ is DEC then, in Definition 9(a), lA B and lA^ B^ have the same Jordan form, J, (it is preserved by strict equivalence), and since lA^ B^ is the linearization of a diagonal system, J 2 Jn;C . ^ lÞ. Thus, the Conversely, LðlÞ has Jordan form J 2 Jn;C implies that there is a strictly isospectral diagonal system Lð ^ ^ corresponding pencils lA B and lA B are isospectral. But then it follows from the Kronecker reduction of regular pencils ([14, Theorem 3.1], for example) that the pencils are strictly equivalent to the same canonical form, and hence to one another. (b) If LðlÞ is a real system which is DER then, in Definition 9(b), lA B and lA^ B^ have the same Jordan form J and, because lA^ B^ is generated by a real diagonal system, J 2 Jn;R , as required. The converse argument is as in (a) but over the real field; i.e. U and V can be taken with real entries [14, Theorem 3.2]. (c) If LðlÞ is an Hermitian system which is DCR then in Definition 9(c), lA B and lA^ B^ have the same Jordan form J and, because lA^ B^ is generated by a real diagonal system, J 2 Jn;R , as required. Conversely, let the Hermitian system LðlÞ have Jordan form J 2 Jn;R . We are to prove that there is a real diagonal system ^ lÞ such that the corresponding linearizations lA B and lA^ B^ are congruent. Lð Let the sign characteristic of LðlÞ be e (i.e. a fixed collection of þ1’s and 1’s associated with the partial multiplicities of the real eigenvalues.3See [8, Theorem 3.7] or [9, Theorem 12.5], for example). Then J 2 Jn;R implies that ^ lÞ. According to (13) the partial multiplicities of the eigenvalues of LðlÞ there are isospectral real diagonal systems Lð ^ (and then of LðlÞ) are either 2 or 1. By Proposition 10.12 in [9]4 we conclude that, for the real semisimple eigenvalues (i.e. those with only linear elementary divisors) the number of signs þ1 is equal to the number of ^ lÞ with distinct real zeros necessarily combine pairs of eigenvalues with signs 1. Furthermore, the diagonal terms of Lð opposite signs. ^ lÞ, multiplication by a diagonal of þ1’s and 1’s and exchanging factors corresponding to Now, given one such Lð semisimple real eigenvalues along the diagonal generates another isospectral diagonal system. Using this freedom, and knowing the signs attached to the real eigenvalues of LðlÞ, corresponding signs can be associated with the real ^ lÞ. In this way an Lð ^ lÞ is determined which is strictly isospectral with LðlÞ (in the terminology eigenvalues of Lð of [13]). ^ lÞ, respectively, and note that each one Now let lA B, lA^ B^ be the usual (Hermitian) linearizations of LðlÞ and Lð inherits both the spectrum and sign characteristic of the parent polynomial. Then it follows from Theorem 6.1 of [14] (attributed to Weierstrass) that the pencils lA B and lA^ B^ have the same canonical forms and are therefore congruent. Thus, LðlÞ is DCR. & We show next how the result of Theorem 10 can be applied to the system in Example 7. Example 8. Consider system LðlÞ in Example 7. The linearization of this system is 2
200
6 100 6 A¼6 4 1
3
2
100
1
0
100
0
0 2
0 0
27 7 7; 05
0
0
2000
6 1000 6 B¼6 4 0 0
3
1000
0
0
2000
0
0 0
1 0
07 7 7. 05 2
The aim is to obtain structure preserving transformations U and V for which UðlA BÞV ¼ lA^ B^ where 2 6 6 A^ ¼ 6 4
227:8131
0
1
0
3
2 6 6 B^ ¼ 6 4
0
22:1869
0
1
0
0
17 7 7; 05
0
1
0
0
3
2290:7119
0
0
0
0
654:8182
0
0
0
1
07 7 7 05
0
0
0
1
3 In Example 1 the eigenvalue l ¼ 1 has a partial multiplicity of order 4 with signature þ1. And in Example 3 L1 ðlÞ and L2 ðlÞ have two eigenvalues: l1 ¼ 1 and l2 ¼ 0. In addition, l1 has two partial multiplicities: 2 and 1. The signature of the eigenvalue l ¼ 1 with partial multiplicity 2 (i.e. the elementary divisor ðl 1Þ2 ) is þ1 for both systems. The signature for the elementary divisors ðl 1Þ is þ1 in system L1 ðlÞ and 1 in system L2 ðlÞ. Finally, the signature of the eigenvalue 0 is 1 in system L1 ðlÞ and þ1 in system L2 ðlÞ. 4
The result is proved here in the case M ¼ I, but extension to nonsingular Hermitian M is not difficult. Or see Proposition 4.2 of [7].
ARTICLE IN PRESS 1142
P. Lancaster, I. Zaballa / Mechanical Systems and Signal Processing 23 (2009) 1134–1144
^ lÞ. This is not an elementary matter but for this example and with the help of is the linearization of the diagonal system Lð MATLAB one can see that the following matrices work (notice that U and V are real): U= 1.0e+03* 0.0099 0.0007 0.0009 0.0003
0.0028 0.0002 0.0003 0.0005
2.1672 0.1878 0.2056 0.0057
0.6016 0.3224 0.0571 0.0112
20.4932 0.9530 216.0165 10.0342
0.0177 0.0296 0.4589 0.8731
0.0943 0.0044 0.9898 0.0450
0.0007 0.0013 0.0022 0
227.8131 0.0000 1.0000 0.0000
0.0000 22.1869 0.0000 1.0000
1.0000 0.0000 0.0000 0.0000
0.0000 1.0000 0.0000 0.0000
1.0e+03 * 2.2907 0.0000 0.0000 0.0000
0.0000 0.6548 0.0000 0.0000
0.0000 0.0000 0.0010 0.0000
0.0000 0.0000 0.0000 0.0010
V=
bU*A*V ans =
bU*B*V ans =
6. Conclusions A complete characterization has been given of n n quadratic systems LðlÞ which can be diagonalized by applying structure preserving transformations (congruence or strict equivalence) to the 2n 2n linearization lA B of the system (see (1)). This raises the familiar question of diagonalizing LðlÞ itself by strict equivalence or congruence. This theory has been reviewed and improved. This paper is part of a more general research program with the goal of parametrization of structure preserving transformations which, in turn, can be used in the design of filters (see [6]). These filters, which transform LðlÞ to an isospectral diagonal form, are to be physically implemented. A natural extension of the theory would be to the diagonalization of n n systems of higher order, r42. To accomplish this in the n-dimensional space of the system itself would be unrealistic for most applications. However, diagonalization of linearizations would be feasible and admit greater flexibility (in terms of Jordan structures) as r increases.
Acknowledgements This project was partially funded by EPSRC (UK) Grant EP/E046290. The authors are duly grateful to the EPSRC and to their partners in this project, S.D. Garvey, U. Prells and A.A. Popov, of the University of Nottingham for constant support and useful advice. The authors are also grateful to a reviewer whose comments led to a significant improvement in exposition. Appendix A. Canonical forms for systems that can be decoupled Given Theorem 6 and the complete descriptions of the admissible Jordan forms Jn;C and Jn;R for systems that can be decoupled, we can obtain the canonical forms for the corresponding linearizations lA B of (1). In particular, since we assume that M is nonsingular, attention can be confined to the tractable cases in which A is nonsingular. The classes of transformations admitted are: (1) Strict equivalence over C. (2) Strict equivalence over R. (3) Congruence over C. (4) Congruence over R. The review of [14] will assist in this task. A.1. Complex systems—no symmetry: strict equivalence over C [14, Theorem 3.1] There is a canonical pencil of the form s M j¼1
ðlI ‘j J ‘j ðlj ÞÞ.
ARTICLE IN PRESS P. Lancaster, I. Zaballa / Mechanical Systems and Signal Processing 23 (2009) 1134–1144
1143
Thus, this form is determined entirely by the Jordan form. The only constraints on the parameters are that ‘j p2 for each j P and sj¼1 ‘j ¼ 2n. A.2. Hermitian systems: strict equivalence over C [14, Theorem 5.1] The sizes of Jordan blocks for the real eigenvalues are not greater than two and for nonreal eigenvalues they have size one. Suppose that there are (a) real eigenvalues a1 ; . . . ; ar with a partial multiplicity equal to one, (b) real eigenvalues arþ1 ; . . . ; as with a partial multiplicity equal to two, and (c) nonreal conjugate pairs msþ1 iosþ1 ; . . . ; mt iot with partial multiplicity one. Then there is a canonical pencil of the form " # " r s t 0 M M M 1 l aj ðl aj Þ l ðmj ioj Þ l aj 0 j¼1
j¼rþ1
l ðmj þ ioj Þ 0
j¼sþ1
# .
A.3. Hermitian systems: congruence over C [14, Theorem 6.1] With the same conventions as Section A.2, there is a canonical pencil of the form " # " # r s t 0 l ðmj þ ioj Þ M M M 1 l aj Zj ðl aj Þ Zj . l ðmj ioj Þ 0 l aj 0 j¼1
j¼rþ1
j¼sþ1
The numbers Z1 ; . . . ; Zs take the values þ1 or 1. They form the sign characteristic of the system. The canonical forms of Cases 2 and 3 differ only in this respect. A.4. Real systems—no symmetry: strict equivalence over R [14, Theorem 3.2] Description of canonical forms over R requires the introduction of two more standard forms. (They will account for the presence of nonreal eigenvalues.) For real numbers, m and oa0 define real Jordan blocks by 2 3 m o 1 0 " # 6 7 m o 0 17 6 o m 7. ; J 2 ðm ioÞ ¼ 6 J 1 ðm ioÞ ¼ 6 0 o m m o7 0 4 5 0 0 o m (The subscripts 1 and 2 refer to the partial multiplicities of eigenvalues of the form m io). Suppose that there are (a) (b) (c) (d)
real eigenvalues a1 ; . . . ; ar with a partial multiplicity equal to one; real eigenvalues arþ1 ; . . . ; as with a partial multiplicity equal to two; nonreal conjugate pairs msþ1 iosþ1 ; . . . ; mt iot with partial multiplicity one; nonreal conjugate pairs mtþ1 iotþ1 ; . . . ; mu iou with partial multiplicity two.
Then there is a canonical pencil of the form " # r s t u M M M M l aj 1 ðl aj Þ ðlI 2 J 1 ðmj ioj ÞÞ ðlI 4 J 2 ðmj ioj ÞÞ. l aj 0 j¼1
j¼rþ1
j¼sþ1
j¼tþ1
A.5. Real-symmetric systems: strict equivalence over R [14, Theorem 9.1] For real numbers m and oa0 define 2
" K 1 ðm ioÞ ¼
#
o m ; m o
0 6 61 K 2 ðm ioÞ ¼ 6 6o 4
1 0
m m o
3
o m m o 7 7 0 0
7. 0 7 5 0
ARTICLE IN PRESS 1144
P. Lancaster, I. Zaballa / Mechanical Systems and Signal Processing 23 (2009) 1134–1144
Then with the same conventions as Case 4, there is a real-symmetric canonical pencil of the form " # r s t u M M M M 1 l aj ðl a j Þ ðlI 2 K 1 ðmj ioj ÞÞ ðlI 4 K 2 ðmj ioj ÞÞ. l aj 0 j¼1
j¼rþ1
j¼sþ1
j¼tþ1
A.6. Real-symmetric systems: congruence over R [14, Theorem 9.2] With the same conventions as Section A.4, there is a real-symmetric canonical pencil of the form " # r s t u M M M M 1 l aj Zj ðl aj Þ Zj ðlI 2 K 1 ðmj ioj ÞÞ ðlI 4 K 2 ðmj ioj ÞÞ. l aj 0 j¼1
j¼rþ1
j¼sþ1
j¼tþ1
The numbers Z1 ; . . . ; Zs take the values þ1 or 1. They form the sign characteristic of the system. The canonical forms of Cases 5 and 6 differ only in this respect. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [13] [14] [15] [16] [17]
T.K. Caughey, M.E.J. O’Kelly, Classical normal modes in damped linear dynamic systems, ASME J. Appl. Mech. 32 (1965) 583–588. M. Chu, N. Del Buono, Total decoupling of general quadratic pencils, Part I: theory, J. Sound Vibr. 309 (2008) 96–111. M. Chu, N. Del Buono, Total decoupling of general quadratic pencils, Part II: structure preserving isospectral flows, J. Sound Vibr. 309 (2008) 112–128. S.G. Garvey, M.I. Friswell, U. Prells, Co-ordinate transformations for second order systems, Part 1: general transformations, J. Sound Vibr. 258 (2002) 885–909. S.G. Garvey, M.I. Friswell, U. Prells, Z. Chen, General isospectral flows for linear dynamic systems, Linear Algebra Appl. 385 (2004) 335–368. S.G. Garvey, Achieving stable diagonalising filters for second order systems, in: Proceedings of the XXVI IMA Conference, February, 2008. I. Gohberg, P. Lancaster, L. Rodman, Spectral analysis of selfadjoint matrix polynomials, Research Paper 419, Department of Mathematics and Statistics, University of Calgary, 1979. I. Gohberg, P. Lancaster, L. Rodman, Spectral analysis of selfadjoint matrix polynomials, Ann. Math. 112 (1980) 33–71. I. Gohberg, P. Lancaster, L. Rodman, Matrix Polynomials, Academic Press, New York, 1982. P. Lancaster, Some questions in the classical theory of vibrating systems, Bull. Poly. Inst. Jassy 17 (1971) 111–112. P. Lancaster, Inverse spectral problems for semisimple damped vibrating systems, SIAM J. Matrix Anal. Appl. 20 (2007) 279–301. P. Lancaster, U. Prells, Isospectral families of high-order systems, Z. Angew. Math. Mech. 87 (2007) 219–234. P. Lancaster, L. Rodman, Canonical forms for Hermitian matrix pairs under strict equivalence and congruence, SIAM Rev. 47 (2005) 407–443. M. Liu, J.M. Wilson, Criterion for decoupling dynamic equations of motion of linear gyroscopic systems, AIAA J. 30 (1992) 2989–2991. F. Ma, T.K. Caughey, Analysis of linear nonconservative vibrations, ASME J. Appl. Mech. 62 (1995) 685–691. K. Meerbergen, Fast frequency response computation for Rayleigh damping, Int. J. Numer. Methods Eng. 73 (1) (2008) 96–106.