Diagonalizable quadratic eigenvalue problems

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Mechanical Systems and Signal Processing

Mechanical Systems and Signal Processing 23 (2009) 1134–1144

0888-32

doi:10.1

� Cor

E-m1 Pa2 Pa

Vasco G

journal homepage: www.elsevier.com/locate/jnlabr/ymssp

Diagonalizable quadratic eigenvalue problems

Peter Lancaster a,1, Ion Zaballa b,�,2

a Department of Mathematics and Statistics, University of Calgary, Calgary, AB, Canada T2N 1N4b Departamento de Matematica Aplicada y EIO, Euskal Herriko Univertsitatea, Apdo. Correos 644, 48080 Bilbao, Spain

a r t i c l e i n f o

Article history:

Received 30 May 2008

Received in revised form

3 November 2008

Accepted 6 November 2008Available online 30 November 2008

Keywords:

Damped systems

Decoupling

Diagonalization

Linearization

70/$ - see front matter & 2008 Published by

016/j.ymssp.2008.11.007

responding author.

ail addresses: [email protected] (P. L

rtially supported by the Natural Sciences and

rtially supported by the Direccion General de

IC07/154-IT-327-07.

a b s t r a c t

A system is defined to be an n� n matrix function LðlÞ ¼ l2M þ lDþK where

M;D;K 2 Cn�n 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.

1. Introduction

A system is defined to be a matrix function Ll ¼ l2M þ lDþK where M;D;K 2 Cn�n and M is nonsingular. If

M;D;K 2 Rn�n then we have a real system. Similarly, if M;D;K are all Hermitian, or all real and symmetric then the systemis said to be Hermitian or real symmetric, respectively. The system is said to be diagonal or decoupled if M;D and K arediagonal 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 samepartial multiplicities. Thus, a system is diagonalizable or, equivalently, it can be decoupled if it admits an isospectraldiagonal 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 fornonsingular matrices U and V . These are well understood but, even so, some improvement in the existing theory are madehere in Section 2. Those relatively simple cases require that, in effect, one of the coefficients M , D, K is expressed in termsof 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 orstrict equivalence transformations to the familiar isospectral ‘‘linearization’’ lA�B of LðlÞ, where

A ¼D M

M 0

� �; B ¼

�K 0

0 M

� �, (1)

Elsevier Ltd.

ancaster), [email protected] (I. Zaballa).

Engineering Research Council of Canada.

Investigacion Cientıfica y Tecnica, Proyecto de Investigacion MTM2007-67812-C02-01 and Gobierno

www.sciencedirect.com/science/journal/ymssp

www.elsevier.com/locate/jnlabr/ymssp

dx.doi.org/10.1016/j.ymssp.2008.11.007

mailto:[email protected]

mailto:[email protected]

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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 confiningattention to those transformations which preserve the block structure of A and B (the so-called structure preservingtransformations). Algorithms of this kind have been studied by Garvey et al. [4,5] and by Chu and Del Buono [2,3], and it isclaimed that ‘‘almost all’’ systems LðlÞ can be diagonalized in this way. Our main result is Theorem 10, in which this claimis 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 keyquestion now is what systems are diagonalizable. The answer is given in Section 3 providing a complete description of theadmissible 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 forillustration. Real application and physical implementation of these concepts are expected in the near future through thedesign of filters (see [6]).

Our analysis depends on detailed knowledge of the theory of reduction of matrix pairs by congruenceor 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-symmetricpencil lA�B by congruence without assuming that the leading coefficient is positive definite, knowledge of thenotion of the sign characteristic of each real eigenvalue is required. (It appears in Sections A.3 and A.6 of the Appendix, forexample, 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, andsimilarly 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 multiplicitiesand its algebraic multiplicity is the sum of the partial multiplicities. For completeness, and for the reader’s convenience, anAppendix is provided giving a summary of canonical forms for linearizations under either congruence or strictequivalence transformations.

2. Diagonalization without linearization

There are many engineering applications in which the system is Hermitian and M is positive definite (M40) and, in theabsence 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 andcomputational literature and are variously known as ‘‘modal’’ or ‘‘Rayleigh’’ damping. See [17], for example, for aninteresting 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 bediagonalized simultaneously then the Caughey–O’Kelly commutativity condition DM�1K ¼ KM�1D, or somethingequivalent, 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 eigenvaluescan arise and, as described above, they have either positive type, negative type, or mixed type. (See [14] orAppendix B of [11], for example.) Here, we first admit semisimple (i.e. nondefective) real eigenvalues with no restrictionon the type.

Lemma 1. Let M;K 2 Cn�n with detMa0, M� ¼M , K� ¼ K. Assume that lM þK is semisimple with all eigenvalues real

and define

L ¼ diag½l1I1; l2I2; . . . ;lsIs�; S ¼ diag½�I1;�I2; . . . ;�Is�, (2)

where the size of the identity matrix Ij 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 Cn�n such that

V �MV ¼ S; V �KV ¼ SL. (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 Ij.

Proof. This is a classical result. It is a special case of Theorem 6.1 of [14]. (Note that there may be repetitionsamong the lj.) &

A result closely analogous to Lemma 1 holds in the case that M;K 2 Rn�n. It is only necessary to use congruenceover R and to replace the unitary matrices Aj by real orthogonal matrices. (This is a special case of Theorem 9.2of [14].) Note carefully that the further condition M40 would ensure that all eigenvalues are real and of positive type,

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P. Lancaster, I. Zaballa / Mechanical Systems and Signal Processing 23 (2009) 1134–11441136

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 generalizedas follows:

Theorem 2. Let the hypotheses of Lemma 1 hold, assume also that all eigenvalues of lM þK have definite type, and that

D� ¼ D. Then there exists a nonsingular U 2 Cn�n such that U�MU , U�DU , and U�KU are diagonal if and only if

DM�1K ¼ KM�1D. If,in addition, M;D;K are real and symmetric, then there is a corresponding U 2 Rn�n.

Proof. We use the notations introduced in Lemma 1. From (3) we obtain

M�1K ¼ ðVSV �ÞðV ��SLV �1Þ ¼ VS2LV �1 ¼ VLV �1, (4)

KM�1 ¼ ðV ��SLV �1ÞðVSV �Þ ¼ V ��SLSV � ¼ V ��LU�. (5)

If also DM�1K ¼ KM�1D then DVLV �1 ¼ V ��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 itfollows that V �DV is block-diagonal (as in (2)). But, as the blocks of V �DV are Hermitian, the unitary blocks of matrix A ofLemma 1 can be chosen to further reduce V �DV to diagonal form.

Conversely, if M0:¼U�MU ;D0:¼U

�DU ;K0 ¼ U�KU are diagonal, it is easily verified that DM�1K ¼ KM�1D.

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 usedabove by strict equivalence transformations. This possibility was investigated by Ma and Caughey [16] and it is interestingthat, 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

LðlÞ ¼ l2 1 0

0 1

� �þ l

0 1

0 0

� �þ

1 0

0 1

� �.

One possibility is to replace the condition that lM þK is semisimple by requiring distinct eigenvalues for this pencil(which is obviously not satisfied by the counter-example).

Lemma 3. Let M;K 2 Cn�n with detMa0, assume that lM þK is semisimple, and write a diagonal matrix of the eigenvalues

of lM þK in the form

L ¼ diag½l1I1; l2I2; . . . ; lsIs�,

where lialj when iaj. Then there is a family of nonsingular matrices U ;V 2 Cn�n such that

UMV ¼ I and UKV ¼ L. (6)

If A ¼ diag½A1; . . . ;As� is nonsingular and Aj has the size of Ij, then U ;V can be replaced by A�1U ;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 strictequivalence transformations of real systems) using Theorem 3.2 of [14], for example.

Theorem 4. Let M;D;K 2 Cn�n with detMa0 and assume that lM þK has n distinct eigenvalues. Then there exist

nonsingular U ;V 2 Cn�n such that UMV ¼ I, and UDV , UKV are diagonal if and only if DM�1K ¼ KM�1D.

Proof. First use Lemma 3 to obtain nonsingular U ;V 2 Cn�n such that UMV ¼ I and UKV ¼ L, a diagonal matrix. Then

M�1K ¼ ðVUÞðU�1LV �1Þ ¼ VLV �1, (7)

KM�1 ¼ ðU�1LV �1ÞðVUÞ ¼ U�1LU . (8)

If also DM�1K ¼ KM�1D then DðVLV �1Þ ¼ ðU�1LUÞD and hence

ðUDV ÞL ¼ LðUDV Þ. (9)

Since L is diagonal with distinct diagonal entries, this implies that UDV is also diagonal, as required.

Conversely, if M0:¼UMV ;D0:¼UDV ;K0 ¼ UKV are diagonal, it is easily verified that DM�1K ¼ KM�1D. &

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If the assumption that lM þK has distinct eigenvalues is relaxed and we assume only that lM þK 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 choiceof matrix A of Lemma 3 to reduce the blocks of UDV by similarity (the strategy proposed in [16]). This will be possibleprovided 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 verystrong commutativity condition must be satisfied. However, as we see now, most systems admit diagonal isospectralsystems; i.e. diagonal systems with the same Jordan form as the original one. As in the Introduction, these systems will becalled diagonalizable and we aim to completely describe them through their Jordan forms. Once a system is known to bediagonalizable, the natural question is how to reduce it to diagonal form. It is shown in Section 5 that this can alwaysbe 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 thecoefficients M;D;K are already diagonal. They can be seen as the ‘‘target’’ systems to which more general systems are to bereduced (whenever possible). The Jordan form for a system LðlÞ is the classical canonical form for A�1B under similaritytransformations 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 � C2n�2n).

A Jordan block with eigenvalue lj and size s is denoted by

JsðljÞ:¼

lj 1 0 � � � 0

0 lj 1 � � � 0

..

. ...

..

.� � � 0 1

0 0 � � � 0 lj

2666666664

3777777775

.

It will be convenient to denote a direct (diagonal) sum of scalars or matrices, x1; . . . ;xk byLk

j¼1 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 beformed in this way. However, a similar construction for Jn;R requires that the 2n numbers consist of, say, r pairs of realnumbers (0prpn) and n� r pairs of (nonreal) conjugate complex numbers. As long as all 2n numbers are distinct thecorresponding Jordan forms will be diagonal, but the situation becomes more complicated if repetitions among theeigenvalues are permitted.

Assume that there exist distinct eigenvalues l1; . . . ; lt 2 C;1ptp2n, and let li have partial multiplicitiesni1X � � �Xni;mg;i40 for each i (forming the ‘‘Segre characteristic’’). Then eigenvalue li has geometric multiplicity mg;ipn

and algebraic multiplicity ma;i ¼Pmg;i

j¼1 nijp2n. Also,

Xti¼1

Xmg;ij¼1

nij ¼ 2n. (10)

Write LðlÞ ¼Ln

i¼1 ½mil2þ dilþ ki�; where

Qni¼1 mia0. Then each diagonal entry has a linearization

lI2 �0 1

�ki=mi �di=mi

" #; i ¼ 1;2; . . . ; t, (11)

and LðlÞ has the tridiagonal linearization lI �A where

A ¼Mni¼1

0 1

�ki=mi �di=mi

" #. (12)

Furthermore, the elementary divisors of lI �A are just the disjoint union of those of (11). Hence,

1pnijp2 for 1pipt; 1pjpmg;i. (13)

For each distinct eigenvalue li, i.e. for i ¼ 1;2; . . . ; t, we define the integers siX0 by writing

nij ¼2 for j ¼ 1;2; . . . ; si;

1 for j ¼ si þ 1; . . . ;mg;i:

((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 (thenumber of quadratic elementary divisors) then the remaining n� p diagonal blocks of A cannot have repeated eigenvalues

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because they are nonderogatory matrices. Thus, in (14) we have

mg;i � sipn� p; i ¼ 1;2; . . . ; t. (15)

We now claim:

Theorem 6. A Jordan matrix J with Segre characteristic fnijgi¼t;j¼mg;ii¼1;j¼1 (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 siX0 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 � s1Xmg;2 � s2X � � �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� 2companion matrices.

The remaining elementary divisors are linear and their eigenvalues can be sorted into distinct pairs, say ðlki ; lkj Þ with

lkialkj . Since the number of elementary divisors of degree two is p:¼s1 þ s2 þ � � � þ st, the number of linear elementary

divisors is 2ðn� pÞ ¼Pt

i¼1 ðmg;i � siÞ. Conditions (15) and (10) ensure that the corresponding eigenvalues can be sorted into

n� p pairs ðlki ; lkj Þ with lkialkj .In other words, the eigenvalues with linear elementary divisors can be organized in two ordered lists, each with n� p

eigenvalues, say

ðapþ1; apþ2; . . . ; anÞ and ðbpþ1; bpþ2; . . . ; bnÞ

with aiabi for i ¼ pþ 1; . . . ;n. For each such i let Bi be the companion matrix of ðl� aiÞðl� biÞ and observe that each Bi is

diagonalizable.

Now the matrix

A ¼ diagðA11; . . . ;A1s1; . . . ;At1; . . . ;Atst ;Bpþ1; . . . ;BnÞ

has J for its Jordan form and, also, lI2n �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;ip2mg;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

LðlÞ ¼ l2 1 0

0 1

� �þ l

2 1

1 2

� �þ

1 1

1 2

� �.

It is easily seen that detLðlÞ ¼ ðlþ 1Þ4 so there is just one eigenvalue, �1, with ma ¼ 4. Also, Lð�1Þ ¼ ½0001�, 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 andthe system is not diagonalizable. The commutativity condition DM�1K ¼ KM�1D may or may not hold (in this case it doesnot hold but there are nondiagonalizable systems for which this condition is satisfied, see Example 2), so Theorem 2 doesnot apply.

Example 2. Let

LðlÞ ¼ l2

1 1 1

1 0 1

1 1 0

264

375þ l

�1 �1 �3=2

�1 0 �2

�3=2 �2 0

264

375þ

0 0 1=2

0 0 1

1=2 1 0

264

375.

In this case detLðlÞ ¼ l2ðl� 1Þ4. The eigenvalue þ1 has ma ¼ 4 and the rank of Lð1Þ is 0. Thus mg ¼ 3 and condition (16) is

fulfilled. For this system, however, JeJn;C because the elementary divisors of LðlÞ are ðl� 1Þ2, ðl� 1Þ, ðl� 1Þ and l2 andcondition (15) is not satisfied.

Although DM�1K ¼ KM�1D 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�1K has the defective eigenvalue zero

with algebraic multiplicity ma ¼ 2 and geometric multiplicity mg ¼ 1.

Example 3. The following two systems:

L1ðlÞ ¼ l2 41 12

12 34

� �þ l

�73 �36

�36 �52

� �þ

32 24

24 18

� �

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and

L2ðlÞ ¼ l2 0 1

1 3

� �þ l

�1 �3

�3 �7

� �þ

1 2

2 4

� �

have the same Jordan structure. In fact, detL1ðlÞ and detL2ðlÞ are scalar multiple of lðl� 1Þ3. So, for the eigenvalue þ1,ma ¼ 3 and since L1ð1Þ ¼ L2ð1Þ ¼ 0, this eigenvalue has geometric multiplicity mg ¼ 2. Actually, for both systems theelementary divisors are: ðl� 1Þ2, ðl� 1Þ and l. Thus, all conditions of Theorem 6 are satisfied and L1ðlÞ, L2ðlÞ and thediagonal system

LðlÞ ¼ l2 1 0

0 1

� �þ l

�1 0

0 �2

� �þ

0 0

0 1

� �

are isospectral.

However, there is an important difference between L1ðlÞ and L2ðlÞ: the mass matrix in L1ðlÞ is positive definite and in

L2ðlÞ it is not. It turns out that L1ðlÞ and LðlÞ are ‘‘strictly’’ isospectral, in the sense of [13] (the sign characteristics are the

same), and L2ðlÞ and LðlÞ are not strictly isospectral.

In addition, for all three systems DM�1K ¼ KM�1D. In particular, Theorem 2 applies for L1ðlÞ; i.e. L1ðlÞ and LðlÞ are

congruent. Also, Theorem 4 applies for L2ðlÞ. That is to say, L2ðlÞ can be reduced to monic diagonal form by real

strict equivalence.

Example 4. Let

LðlÞ ¼ l2 5 �2

�2 1

� �þ l

�319 2666

126 �1053

� �þ

0 319

0 �126

� �.

For this system detLðlÞ ¼ l2ðl2� 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,

LðlÞ ¼ l2 2 0

0 1

� �þ

0 0

0 �1

� �.

However, although lM þK has distinct eigenvalues, DM�1KaKM�1D and so,by Theorem 4, LðlÞ cannot be decoupled bystrict equivalence.

4. Jordan forms for real diagonal systems

Now consider Jordan canonical forms for real diagonal systems.

Theorem 7. A Jordan matrix J with Segre characteristic fnijgi¼t;j¼mg;ii¼1;j¼1 (defined as above) is in Jn;R if and only if there is an n0,

0pn0pn, such that J ¼ diagðJn0; Jn�n0

Þ for Jordan matrices Jn0; Jn�no with sðJn0

Þ � R and sðJn�n0Þ \R ¼ ; and:

(a)
conditions (10), (13) and (15) (with n replaced by n0) hold for Jn0and
(b)
sðJn�n0Þ 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 Jn0simply does not appear, and if

the system (and hence J) has no nonreal eigenvalues then n0 ¼ n and Jn�n0does not appear. To illustrate, in the case of

Example 4, n0 ¼ n and so Jn�n0does not appear and

J ¼ Jn0¼

0 1 0 0

0 0 0 0

0 0 1 0

0 0 0 �1

26664

37775.

Proof. If LðlÞ is a real diagonal system and l0 is an eigenvalue with l0al0, then unit co-ordinate vectors ej can be chosenas 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 Jn�n0satisfying 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. &

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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-symmetricsystem may have nonreal eigenvalues with nonlinear elementary divisors. Thus,

LðlÞ:¼1 0

0 1

� �l2þ

1ffiffiffi3p

=2ffiffiffi3p

=2 2

" #lþ

1 0

0 4

� �(17)

has eigenvalues 14ð�3� i

ffiffiffiffiffiffi23pÞ with elementary divisors of degree two.

Example 6. The following system, however, satisfies the conditions of Theorem 7:

LðlÞ ¼3=2 �1=2

�1=2 3=2

" #l2þ�3 5

5 �3

� �lþ

11=2 �9=2

�9=2 11=2

" #.

In fact, its eigenvalues are: l1 ¼ �1, l2 ¼ 2þ i and l3 ¼ 2� i. Since the eigenvalue �1 has partial multiplicity (2)(i.e. ma ¼ 2 and mg ¼ 1), its Jordan form is in Jn;R. A diagonal isospectral system is

LðlÞ ¼ l2 1 0

0 1

� �þ l

2 0

0 �4

� �þ

1 0

0 5

� �.

Furthermore, it is easily verified that DM�1K ¼ KM�1D. Since M is positive definite, Theorem 2 applies and LðlÞ can bedecoupled 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:

LðlÞ ¼ l2 1 0

0 2

� �þ l

200 �100

�100 100

� �þ

2000 �1000

�1000 2000

� �.

This is not a proportionally damped system and Theorems 2 and 4 show that LðlÞ cannot be reduced to diagonal form byeither congruence or strict equivalence. However, the eigenvalues of LðlÞ are all distinct: �217:2699, �10:5432,�11:0935þ 23:0598i, �11:0935� 23:0598i. By Corollary 8, the Jordan form of LðlÞ is in Jn;R. A diagonal system with thesame Jordan form is

LðlÞ ¼ l2 1 0

0 1

� �þ l

227:8131 0

0 22:1869

� �þ

2290:7119 0

0 654:8182

� �.

5. Linearization and diagonalizable systems

Given a system LðlÞ, we now consider the generation of an isospectral diagonal system (when one exists) by theapplication 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 aredefined 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 C2n�2n

such that

UðlA�BÞV ¼ lA� B,

where lA� B is the linearization of a (generally complex) diagonal system LðlÞ ¼ l2M þ lDþ K.

(b) A real system is DER if there exist nonsingular U ;V 2 C2n�2n such that

UðlA�BÞV ¼ lA� B,

where lA� B is the linearization of a real diagonal system LðlÞ ¼ l2M þ lDþ K.

(c) A system is DCR (diagonalizable by congruence) if there exists a nonsingular U 2 C2n�2n such that

UðlA�BÞU� ¼ lA� B,

where lA� B is the linearization of a real diagonal system LðlÞ ¼ l2M þ lDþ K.

ARTICLE IN PRESS


Notice that, in case (c), if the system is Hermitian (or real and symmetric), then so are A; B and, in particular, because Kand M are diagonal, so is B. The following result now seems natural:

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 preservedby strict equivalence), and since lA� B is the linearization of a diagonal system, J 2 Jn;C.

Conversely, LðlÞ has Jordan form J 2 Jn;C implies that there is a strictly isospectral diagonal system LðlÞ. Thus, the

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

LðlÞ such that the corresponding linearizations lA�B and lA� B are congruent.

Let the sign characteristic of LðlÞ be e (i.e. a fixed collection of þ1’s and �1’s associated with the partial multi-

plicities of the real eigenvalues.3See [8, Theorem 3.7] or [9, Theorem 12.5], for example). Then J 2 Jn;R implies that

there are isospectral real diagonal systems LðlÞ. According to (13) the partial multiplicities of the eigenvalues of LðlÞ(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

signs �1. Furthermore, the diagonal terms of LðlÞ with distinct real zeros necessarily combine pairs of eigenvalues with

opposite signs.

Now, given one such LðlÞ, multiplication by a diagonal of þ1’s and �1’s and exchanging factors corresponding to

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

eigenvalues of LðlÞ. In this way an LðlÞ is determined which is strictly isospectral with LðlÞ (in the terminology

of [13]).

Now let lA�B, lA� B be the usual (Hermitian) linearizations of LðlÞ and LðlÞ, respectively, and note that each one

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

A ¼

200 �100 1 0

�100 100 0 2

1 0 0 0

0 2 0 0

26664

37775; B ¼

�2000 1000 0 0

1000 �2000 0 0

0 0 1 0

0 0 0 2

26664

37775.

The aim is to obtain structure preserving transformations U and V for which UðlA�BÞV ¼ lA� B where

A ¼

227:8131 0 1 0

0 22:1869 0 1

1 0 0 0

0 1 0 0

26664

37775; B ¼

�2290:7119 0 0 0

0 �654:8182 0 0

0 0 1 0

0 0 0 1

26664

37775

3 In Example 1 the eigenvalue l ¼ �1 has a partial multiplicity of order 4 with signature þ1. And in Example 3 L1ðlÞ and L2ðlÞ 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 L1ðlÞ and �1 in system L2ðlÞ. Finally,

the signature of the eigenvalue 0 is �1 in system L1ðlÞ and þ1 in system L2ðlÞ.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


is the linearization of the diagonal system LðlÞ. This is not an elementary matter but for this example and with the help ofMATLAB one can see that the following matrices work (notice that U and V are real):

U =

1.0e+03*

0.0099
�0.0028 �2.1672 0.6016
0.0007
�0.0002 �0.1878 �0.3224
0.0009
�0.0003 �0.2056 0.0571
0.0003
0.0005 �0.0057 �0.0112
V =

� 20.4932
�0.0177 �0.0943 �0.0007
0.9530
�0.0296 0.0044 �0.0013
216.0165
0.4589 0.9898 �0.0022
� 10.0342
0.8731 �0.0450 0
bU*A*V ans =

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
bU*B*V ans =

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
6. Conclusions

A complete characterization has been given of n� n quadratic systems LðlÞ which can be diagonalized by applyingstructure 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 hasbeen reviewed and improved.

This paper is part of a more general research program with the goal of parametrization of structure preservingtransformations which, in turn, can be used in the design of filters (see [6]). These filters, which transform LðlÞ to anisospectral 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 accomplishthis in the n-dimensional space of the system itself would be unrealistic for most applications. However, diagonalization oflinearizations 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 totheir partners in this project, S.D. Garvey, U. Prells and A.A. Popov, of the University of Nottingham for constant support anduseful 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 bedecoupled, we can obtain the canonical forms for the corresponding linearizations lA�B of (1). In particular, since weassume 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) Congruenceover 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

Msj¼1

ðlI‘j � J‘j ðljÞÞ.

ARTICLE IN PRESS


Thus, this form is determined entirely by the Jordan form. The only constraints on the parameters are that ‘jp2 for each j

andPs

j¼1 ‘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 sizeone. 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

Mrj¼1

ðl� ajÞ Msj¼rþ1

1 l� ajl� aj 0

" #Mtj¼sþ1

0 l� ðmj þ iojÞ

l� ðmj � iojÞ 0

" #.

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

Mrj¼1

Zjðl� ajÞ Msj¼rþ1

Zj1 l� aj

l� aj 0

" #Mtj¼sþ1

0 l� ðmj þ iojÞ

l� ðmj � iojÞ 0

" #.

The numbers Z1; . . . ;Zs take the values þ1 or �1. They form the sign characteristic of the system. The canonical forms ofCases 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 thepresence of nonreal eigenvalues.) For real numbers, m and oa0 define real Jordan blocks by

J1ðm� ioÞ ¼m o�o m

" #; J2ðm� ioÞ ¼

m o 1 0

�o m 0 1

0 0 m o0 0 �o m

266664

377775.

(The subscripts 1 and 2 refer to the partial multiplicities of eigenvalues of the form m� io).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; (c) nonreal conjugate pairs msþ1 � iosþ1; . . . ;mt � iot with partial multiplicity one; (d) nonreal conjugate pairs mtþ1 � iotþ1; . . . ;mu � iou with partial multiplicity two.
Then there is a canonical pencil of the form

Mrj¼1


l� aj 1

0 l� aj

" #Mtj¼sþ1

ðlI2 � J1ðmj � iojÞÞ Muj¼tþ1

ðlI4 � J2ðmj � iojÞÞ.

A.5. Real-symmetric systems: strict equivalence over R [14, Theorem 9.1]

For real numbers m and oa0 define

K1ðm� ioÞ ¼o mm �o

" #; K2ðm� ioÞ ¼

0 1 o m1 0 m �oo m 0 0

m �o 0 0

266664

377775.

ARTICLE IN PRESS


Then with the same conventions as Case 4, there is a real-symmetric canonical pencil of the form

Mrj¼1


1 l� ajl� aj 0

" #Mtj¼sþ1

ðlI2 �K1ðmj � iojÞÞ Muj¼tþ1

ðlI4 �K2ðmj � iojÞÞ.

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

Mrj¼1

Zjðl� ajÞ Msj¼rþ1

Zj1 l� aj

l� aj 0

" #Mtj¼sþ1

ðlI2 �K1ðmj � iojÞÞ Muj¼tþ1

ðlI4 �K2ðmj � iojÞÞ.

The numbers Z1; . . . ;Zs take the values þ1 or �1. They form the sign characteristic of the system. The canonical forms ofCases 5 and 6 differ only in this respect.

References

[1] T.K. Caughey, M.E.J. O’Kelly, Classical normal modes in damped linear dynamic systems, ASME J. Appl. Mech. 32 (1965) 583–588.[2] M. Chu, N. Del Buono, Total decoupling of general quadratic pencils, Part I: theory, J. Sound Vibr. 309 (2008) 96–111.[3] M. Chu, N. Del Buono, Total decoupling of general quadratic pencils, Part II: structure preserving isospectral flows, J. Sound Vibr. 309 (2008) 112–128.[4] 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.[5] S.G. Garvey, M.I. Friswell, U. Prells, Z. Chen, General isospectral flows for linear dynamic systems, Linear Algebra Appl. 385 (2004) 335–368.[6] S.G. Garvey, Achieving stable diagonalising filters for second order systems, in: Proceedings of the XXVI IMA Conference, February, 2008.[7] I. Gohberg, P. Lancaster, L. Rodman, Spectral analysis of selfadjoint matrix polynomials, Research Paper 419, Department of Mathematics and

Statistics, University of Calgary, 1979.[8] I. Gohberg, P. Lancaster, L. Rodman, Spectral analysis of selfadjoint matrix polynomials, Ann. Math. 112 (1980) 33–71.[9] I. Gohberg, P. Lancaster, L. Rodman, Matrix Polynomials, Academic Press, New York, 1982.

[10] P. Lancaster, Some questions in the classical theory of vibrating systems, Bull. Poly. Inst. Jassy 17 (1971) 111–112.[11] P. Lancaster, Inverse spectral problems for semisimple damped vibrating systems, SIAM J. Matrix Anal. Appl. 20 (2007) 279–301.[13] P. Lancaster, U. Prells, Isospectral families of high-order systems, Z. Angew. Math. Mech. 87 (2007) 219–234.[14] P. Lancaster, L. Rodman, Canonical forms for Hermitian matrix pairs under strict equivalence and congruence, SIAM Rev. 47 (2005) 407–443.[15] M. Liu, J.M. Wilson, Criterion for decoupling dynamic equations of motion of linear gyroscopic systems, AIAA J. 30 (1992) 2989–2991.[16] F. Ma, T.K. Caughey, Analysis of linear nonconservative vibrations, ASME J. Appl. Mech. 62 (1995) 685–691.[17] K. Meerbergen, Fast frequency response computation for Rayleigh damping, Int. J. Numer. Methods Eng. 73 (1) (2008) 96–106.

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