on solvents of matrix polynomials

conceptinfinite

a greatnomial

thoughients

is wellentalave a

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Applied Numerical Mathematics 47 (2003) 197–208www.elsevier.com/locate/apnum

On solvents of matrix polynomials✩

Edgar Pereira

Departamento de Informática, Universidade da Beira Inteiror, 6200 Covilhã, Portugal

Abstract

First, the basic theory of matrix polynomials and the construction of solvents are reviewed in terms of theof eigenpair. Then a study of the number of solvents is done. Conditions for the number of solvents to beare stated and for a finite (positive) number the minimum and the maximum are estimated. 2003 IMACS. Published by Elsevier B.V. All rights reserved.

Keywords:Matrix polynomials; Solvents

1. Introduction

The study of solutions of polynomial equations dates back about 5 thousand years and hadinfluence on the development of modern mathematics [13]. On the other hand, the interest in polyequations involving matrices is relatively recent, the first references point to the 19th century. AlCaley was the first to study matrix equations, the study of polynomials with matricial coefficprobably started with Sylvester (see [14,15]). The number of solutions of a complex polynomialknown as it is well known that the work to get this seemingly simple result, namely the fundamTheorem of Algebra, has been long and hard. For matrix polynomials, until now, we do not hreasonable characterization of the number of solutions. In this work we investigate the procesconstruction of such solutions and obtain some new results concerning their number. LetA1, . . . ,Am ben× n complex matrices, that is,Ai ∈ C

n×n, i = 1, . . . ,m. The expression

P(X)=Xm +A1Xm−1 + · · · +Am (1)

denotes a monic (right) matrix polynomial of degreem the in unknownX. If the unknown isλ ∈ C,thenP(λ) is a matrix polynomial inλ or aλ-matrix. If X ∈ C

n×n, the polynomial equationP(X) = 0is sometimes called a unilateral matrix equation. A matrix solutionX1 ∈ C

n×n of P(X), such that

✩ This work was partially supported by ISR Pólo de Coimbra.E-mail address:[email protected] (E. Pereira).

0168-9274/$30.00 2003 IMACS. Published by Elsevier B.V. All rights reserved.doi:10.1016/S0168-9274(03)00058-8

198 E. Pereira / Applied Numerical Mathematics 47 (2003) 197–208

P(X1) = 0, is a (right) solvent. For each matrix polynomialP(X) there is a matrix associated withits coefficients, the block companion matrix

t

he

ing

Jordan

e

ofhe

e [7,

C =

0n In . . . 0n...

. . .

In−Am −Am−1 . . . −A1

, (2)

where 0n andIn are the null matrix and the identity matrix of ordern, respectively.

Definition 1. LetP(λ) be a matrix polynomial inλ. If α ∈ C is such that det(P (α))= 0, then we say thaα is a latent root or an eigenvalue ofP(λ). If a nonzerov ∈ C

n is such thatP(α)v = 0 (vector), then wesay thatv is a (right) latent vector or a (right) eigenvector ofP(λ), corresponding to the eigenvalueα.Moreover, if P(λ) = (λIn − A) is of degree 1, thenv is a (right) eigenvector, corresponding to teigenvalueα of the complex matrixA.

Definition 2. Given a matrix polynomialP(λ) of degreem, and an eigenvalueα ∈ C such thatdet(P (α)) = 0, let P (j) denote thej th derivative ofP(λ). If k vectorsv1, . . . , vk ∈ C

n, with v1 �= 0,not necessarily distinct or linearly independent, satisfy

P(α)vi + 1

1!P(1)(α)vi−1 + 1

2!P(2)(α)vi−2 + · · · + 1

m!P(m)(α)vi−m = 0, i = 1, . . . , k, (3)

with vl = 0 if l � 0, we say thatv1, . . . , vk is a Jordan chain, of lengthk, of P(λ) corresponding tothe eigenvalueα. We also say thatv1, . . . , vk are generalized eigenvectors ofP(λ) corresponding toα.Moreover, ifP(λ)= (λI −A) is of degree 1, thenv1, . . . , vk are generalized eigenvectors, correspondto the eigenvalueα of the complex matrixA.

Next we extend the concept of eigenpair, a pair eigenvector-eigenvalue, for the case in whichblocks are of order greater than unity.

Definition 3. Given a matrix polynomial inλ, P(λ), and the matrixV of dimensionn× k,V = [v1 . . . vk ],

wherev1, . . . , vk are generalized eigenvectors or a Jordan chain ofP(λ) corresponding to the eigenvaluα, we say that the pair

(V , J ) , (4)

(whereJ is a Jordan block of sizek, with the eigenvalueα in the main diagonal) is an eigenpairP(λ). If k = 1, then(v1, α) is an eigenpair ofP(λ), wherev1 is an eigenvector corresponding to teigenvalueα. And if the degree ofP(λ) is equal to 1, that isP(λ)= (λIn −A), then(V , J ) (or (v1, α))is an eigenpair of the complex matrixA.

The classical study of solvents of matrix polynomials uses the theory of divisibility (seSections 4.1 and 4.2]), from which we have that, ifX1 is a solvent ofP(X), then(λIn −X1) is a lineardivisor ofP(λ), and hence the eigenvalues, including multiplicities, ofX1 are also eigenvalues ofP(λ).For the explicit computations of solvents, the basic approach is the search for a matrix of the form

QJXQ−1, (5)

E. Pereira / Applied Numerical Mathematics 47 (2003) 197–208 199

whereJX having eigenvalues ofP(λ) is in the Jordan normal form. Mac Duffee and Gantmacher suggestthe search for an arbitrary nonsingular matrixQ satisfying

ssarysary[2] thehere wesize theheing thee.ill alsos. Thee numberimum

ncept of

on

of

rix.

QJmX +A1QJm−1X + · · · +AmQ= 0

(see [12, p. 95] and [5, p. 230]). In fact, this matrix is not so arbitrary. In an earlier work giving nececonditions, Roth [15] stated that the columns of the matrixQ have to verify condition (3). Nowadaywe define Jordan chains forP(λ) using this condition (Definition 2). In [9, p. 53] and in [4], necessand sufficient conditions are given for (5) to be a solvent in the diagonalizable case, while ingeneral case is presented. Our work has two main contributions. The first one is in Section 2 wrewrite the basic theory of solvents using the extension of the concept of eigenpair, to emphafact that not only the eigenvalues of the solvent are eigenvalues ofP(λ) but are also the eigenpairs. Tnovelty here will be that we will obtain these eigenpairs in a simple and comprehensive way ussimilarity matrix of the block companion matrixC. This will be explained carefully in the first examplIn the following examples we will use the same technique but present only the eigenpairs. We wshow that this similarity matrix can be used to construct matrix polynomials with desired eigenpairsecond contribution appears in Section 3 where we present some theorems to determine when thof solvents of a matrix polynomial is either infinite or finite. For the case of a finite number, the minand the maximum will be estimated.

2. Basic theory

We develop here a basic theory on the existence and construction of solvents using the coeigenpair. In addition to the indicated references we use the following sources [15,9,4,6,7,11].

Theorem 1 [8, p. 146].LetP(X) be a matrix polynomial and letC be the associated block companimatrix, then

det(Iλ−C)= det(P(λ)

). (6)

Corollary 1. The eigenvalues and corresponding partial multiplicities are common toC andP(λ).

Definition 4. Let P(λ), (V1, J1), (V2, J2), . . . , (Vl, Jl) be l eigenpairs ofP(λ). If

diag(J1, . . . , Jl)= JC,where diag(J1, . . . , Jl) is a diagonal matrix of ordermn, with J1, . . . , Jl in the main diagonal, andJC isthe Jordan normal form of the block companion matrixC, we say that(V1, J1), (V2, J2), . . . , (Vl, Jl) is acomplete system of eigenpairs ofP(λ) (if P(λ)= (λIn −A) we also say that it is a complete systemeigenpairs ofA).

Theorem 2 [10]. LetP(X) be a matrix polynomial and letC be the associated block companion matLetS be the similarity matrix ofC, that is,S is such that

C = SJCS−1.


ThenS has the form

alue

S =V1 V2 . . . VlV1J1 V2J2 VlJl...

......

V1Jm−11 V2J

m−12 . . . VlJ

m−1l

, (7)

where (V1, J1), (V2, J2), . . . , (Vl, Jl) is a complete system of eigenpairs ofP(λ), that is diag(J1, . . . ,

Jl)= JC .

Remark 1. If C is diagonalizable, we have

S =

v1 v2 . . . vmnα1v1 α2v2 . . . αmnvmn...

......

αm−11 v1 αm−1

2 v2 . . . αm−1mn vmn

, (8)

wherevi, i = 1, . . . ,mn, are eigenvectors ofP(λ) corresponding to the eigenvaluesαi of P(λ) and alsoof C.

Theorem 3 [6]. The pair of matrices

(V , J ),

is an eigenpair of a matrix polynomialP(λ) if and only if

V Jm +A1V Jm−1 + · · · +AmV = 0.

Theorem 3 is originally formulated in terms of Jordan pairs. A Jordan pair(W1, J1) has a structuresimilar to an eigenpair, withJ1 being the direct sum of all Jordan blocks relating to a common eigenvα1 of P(λ) andW1 having the respective Jordan chains as its columns.

Corollary 2. Given a pair of matrices(T , J0), whereT = [V1 . . . Vl ] is a matrix of dimensionn×p,andJ0 = diag(J1, . . . , Jl) is of orderp, then(Vi, Ji), i = 1, . . . , l, are eigenpairs ofP(λ) if and only if

T Jm0 +A1T Jm−10 + · · · +AmT = 0. (9)

Theorem 4. If the matrix T = [V1 . . . Vp ] is a nonsingular matrix of ordern and J0 = diag(J1,

. . . , Jl) is also of ordern, then(Vi, Ji), i = 1, . . . , l, are eigenpairs of a matrix polynomial inλ, P(λ), ifand only if

X1 = T J0T−1, (10)

is a solvent ofP(X).

Proof. By Corollary 2, we have successively

0= T Jm0 +A1T Jm−10 + · · · +Am−1T J

10 +AmT

= (T Jm0 +A1T J

m−10 + · · · +Am−1T J

10 +AmT

)T −1


= T Jm0 T −1 +A1T Jm−10 T −1 + · · · +Am−1T J

10T

−1 +AmT T −1( −1)m ( −1

)m−1 ( −1)

e

de

ts

= T J0T +A1 T J0T + · · · +Am−1 T J0T +Am= (X1)

m +A1(X1)m−1 + · · · +Am−1(X1)+Am = P(X1). ✷

Definition 5. LetX1, . . . ,Xm bem solvents of a matrix polynomialP(X), and(Vi1, Ji1), . . . , (Vili , Jili )a complete system of eigenpairs of eachXi, i = 1, . . . ,m (whereli is the number of Jordan blocks of thJordan normal form ofXi). If (V11, J11), . . . , (V1l1, J1l1), . . . , (Vm1, Jm1), . . . , (Vmlm, Jmlm) is a completesystem of eigenpairs ofP(λ), then we say thatX1, . . . ,Xm is a complete set of solvents ofP(X).

The next result follows easily from Theorem 2.

Theorem 5. If X1, . . . ,Xm is a complete set of solvents ofP(X) then the respective block Vandermonmatrix

V (X1, . . . ,Xm)=

In . . . InX1 . . . Xm...

...

Xm−11 . . . Xm−1

m

,

is nonsingular.

Theorem 6 [11, p. 524].Let P(X) be a matrix polynomial. IfX1, . . . ,Xm is a complete set of solvenof P(X) then

C = V (X1, . . . ,Xm) diag(X1, . . . ,Xm)V (X1, . . . ,Xm)−1. (11)

We illustrate the theory above with the following example.

Example 1. Consider the matrix polynomial of degree 2

P(X)=X2 +A1X+A2,

with coefficients given by

A1 =[−213 −180

240 203

], A2 =

[692 540

−792 −618

],

wherem= 2 andn= 2. The block companion matrix ofP(X) is

C =[

02 I2

−A2 −A1

],

with C = S−1JCS, where

S =

3 2 0 5−4 −3 1 −43 4 0 20

−4 −6 3 −16

, JC =

1 0 0 00 2 0 00 0 3 00 0 0 4

.


FromS (and JC) we obtain a complete system of eigenpairs ofP(λ):([ ] ) ([ ] ) ([ ] ) ([ ] )]).

is

used to

3−4

,1 ,2

−3,2 ,

01,3 ,

5−4

,4 .

We have that any subset ofn= 2 eigenvectors ofP(λ) is linearly independent (Haar condition, [3, p. 74Hence, with any 2 eigenvectors it is possible to construct a nonsingular matrixT . From Theorem 4it follows thatX0 = T J0T

−1, for a suitableJ0, is a solvent ofP(X). The total number of solvents(m

n

)= (42

)= 6. In addition,

X1 =[

3 2−4 −3

][1 00 2

][3 2

−4 −3

]−1

=[−7 −6

12 10

],

X2 =[

0 51 −4

][3 00 4

][0 51 −4

]−1

=[

4 0−4/5 3

],

is a complete set of solvents ofP(X), sinceV (X1,X2) is nonsingular.

Now we show that the existence of a nonsingular matrix with the format of the matrixS impliesthe existence of a block companion matrix associated to a matrix polynomial. This fact can beconstruct matrix polynomials with predetermined Jordan pairs, or even with certain solvents.

Theorem 7. Let

S =

V1 V2 . . . VlV1J1 V2J2 VlJl...

......

V1Jm−11 V2J

m−12 . . . VlJ

m−1l

, (12)

be a matrix of ordermn, whereVi , i = 1,2, . . . , l, are n × ki matrices andJi is a Jordan block ofsizeki with αi on the diagonal, fori = 1,2, . . . , l, and

∑li=1 ki = mn. If S is nonsingular thenS is a

similarity matrix of a block companion matrix associated with a matrix polynomialP(X). Furthermore,(V1, J1), (V2, J2), . . . , (Vl, Jl) is a complete system of eigenpairs ofP(λ).

Proof. LetA be a matrix of ordermn such that

AS = S diag(J1, J2, . . . , Jl), (13)

where the matrixS is nonsingular. ThereforeA exists, is unique and similar to diag(J1, J2, . . . , Jl). Sowe only have to show thatA is a block companion matrix. Let us write

A=

A11 A12 . . . A1m

A21 A22 . . . A2m...

...

Am1 Am2 . . . Amm

,

where the blocksAij , i = 1, . . . ,m, j = 1, . . . ,m, are of ordern. Now

S diag(J1, J2, . . . , Jl)=

V1J1 V2J2 . . . VlJlV1J

21 V2J

22 VlJ

2l

......

...

V1Jm1 V2J

m2 . . . VlJ

ml

.


Expanding both sides of Eq. (13) we get, for the first row of blocks the system,m−1

ent

atrixhe

a

.

of

. This

A11V1 +A12V1J1 + · · · +A1mV1J1 = V1J1,

A11V2 +A12V2J2 + · · · +A1mV2Jm−12 = V2J2,

...

A11Vl +A12VlJl + · · · +A1mVlJm−1l = VlJl,

which has the solution

A11 = 0n, A12 = In, A13 = 0n, . . . , A1m = 0n.

Continuing the process for the second row of blocks, we get

A21 = 0n, A22 = 0n, A23 = In, A24 = 0n, . . . , A2m = 0n,

and so on, until the(m− 1)th row of blocks, that implies

A(m−1)1 = 0n, A(m−1)2 = 0n, . . . , A(m−1)(m−1) = 0n, A(m−1)m = In.Thus we obtain

A=

0n In . . . 0n...

. . .

0n InAm1 Am2 . . . Amm

.

Hence,A is the block companion matrix of the matrix polynomial

P(X)=Xm + (−Amm)Xm−1 + · · · + (−Am1),

and, by Theorem 2,(V1, J1), (V2, J2), . . . , (Vl, Jl) is a complete system of eigenpairs ofP(λ). ✷

3. Number of solvents

A direct consequence of Theorem 4 is that, for each nonsingular matrix of ordern constructed withthe leading vectors of the Jordan chains ofP(λ), if we take the respective eigenpairs then one solvof P(X) is obtained. Therefore the total number of solvents ofP(X) will be the number of suchmatrices of Jordan chains ofP(λ). Furthermore, these Jordan chains are the columns of the submof dimensionn×mn on the top of the similarity matrixS of C (see Theorem 2). Thus, our study of tnumber of solvents ofP(X) will be done in terms of the block companion matrixC, using its similaritymatrix S. We start with the case where the block companion matrixC is diagonalizable, in which, asconsequence of Corollary 1, all Jordan chains of the matrix polynomial inλ, P(λ) associated withC areof size 1. ThusP(λ) hasmn eigenvectors (up to a constant multiple) corresponding to itsmn eigenvalues

Lemma 1 [4]. Let A be a matrix of ordermn. If A is nonsingular then there exists a permutationcolumns such that the new matrix hasm diagonal blocks of ordern, each of them being nonsingular.

The following Theorem gives us a preliminary estimate of the minimum number of solventsestimate will be improved later on in this section.


Theorem 8. LetP(X) be a matrix polynomial and letC be the associated block companion matrix. IfCis diagonalizable, thenP(X) has at leastm solvents.

lngat the

f

Proof. Since we have the equalityC = SJCS−1, with S nonsingular andJC diagonal, the equality wilremain valid if the same permutation of columns is applied toS and JC . From Lemma 1, we casuppose that them diagonal blocksSi of order n, i = 1, . . . ,m, of S are nonsingular. Considerinthat an eigenvector multiplied by a constant is still an eigenvector, we have from Remark 1 thcolumns ofSi are eigenvectors ofP(λ). Hence, eachXi = SiJCiS−1

i , i = 1, . . . ,m, is a solvent (whereJCi , i = 1, . . . ,m, are diagonal blocks of ordern of JC ). Thus we have at leastm solvents. ✷Corollary 3. For every eigenvalueαi ofC there exists a solventX1, such thatαi is an eigenvalue ofX1.

The following two examples illustrate that, if the matrixC is not diagonalizable, the associatedP(X)can have less thanm solvents or even no solvents at all.

Example 2. Consider the matrix polynomial

P(X)=X2 +A1X+A2,


A1 =[−282/49 318/49

43/49 −355/49

], A2 =

[298/49 −776/49−51/49 584/49

],

wherem= 2 andn= 2. A complete system of eigenpairs ofP(λ) is

(V1, J1)=([−2

1

],7

), (V2, J2)=

([−2 1 31 3 4

],

[2 1 00 2 10 0 2

]).

There is only one nonsingular matrix constructed with the leading vectors of the Jordan chains oP(λ)

(leading columns ofV1 andV2):

T =[−2 1

1 3

].

Thus the unique solvent ofP(X) is

X1 =[−2 1

1 3

][2 10 2

][−2 11 3

]−1

=[

12/7 −4/71/7 16/7

].


P(X)=X2 +A1X+A2,


A1 =[−98/25 108/25 −112/25

4/5 −24/5 −4/522/25 38/25 −182/25

], A2 =

[ 89/25 −294/25 316/25−7/5 42/5 −8/5

−46/25 −59/25 251/25

],

wherem= 2 andn= 3. A complete system of eigenpairs ofP(λ) is


(V1, J1)=([2 2

1 1

],

[2 10 2

]),

er of

f

tat

t

stronger

1 1

(V2, J2)=[−2 −2 5 3

2 2 −1 −11 1 1 1

],

3 1 0 00 3 1 00 0 3 10 0 0 3

.

It can be verified that there are no nonsingular matrices of order 3 with the leading columns ofV1 andV2.Therefore,P(X) has no solvents at all.

Next we see that ifC is diagonalizable with at least one multiple eigenvalue, then the numbsolvents is infinite. We use the fact that the partial multiplicities of the eigenvalues are the same inC andin P(λ) (Corollary 1).

Theorem 9. LetP(X) be a matrix polynomial and letC be the associated block companion matrix. ICis diagonalizable and at least one of its eigenvalues has geometric multiplicity greater than1, thenP(X)has infinitely many solvents.

Proof. Let (v1, α1), (v2, α2), . . . , (vmn,αmn) be a complete system of eigenpairs ofP(λ). Suppose thaα1 is the eigenvalue with geometric multiplicity greater than 1. Now, from Corollary 3, consider thX1

is a solvent of whichα1 is an eigenvalue. For simplicity, suppose

X1 = [v1 v2 . . . vn ]

α1 0 . . . 00 α2 . . . 0...

.... . . 0

0 0 . . . αn

[v1 v2 . . . vn ]−1.

Let vk be another eigenvector corresponding toα1. By hypothesis,v1 andvk are linearly independenand span a subspaceS2×1

1 ⊂ Cn×1 of dimension 2. It is clear that the intersection ofS

2×11 with the

subspace spanned byv2, . . . , vn has at most dimension 1, since matrix[v1 v2 . . . vn ] is of rankn.Then there exists an infinite number of eigenvectorsvi = βv1 + γ vk, for nonzero complexβ,γ , with[vi v2 . . . vn ] being nonsingular. Hence, there are infinitely many solvents

Yi = [vi v2 . . . vn ]

α1 0 . . . 00 α2 . . . 0...

.... . . 0

0 0 . . . αn

[vi v2 . . . vn ]−1. ✷

In the case where the block companion matrix is not supposed to be diagonalizable, we needconditions for the occurrence of an infinitude of solvents.

Theorem 10. Let P(X) be a matrix polynomial,C the associated block companion matrix, andX1 asolvent ofP(X). If one eigenvalue ofX1 has the geometric multiplicity greater inC than inX1, thenP(X) has infinitely many solvents.


Proof. Let (V1, J1), (V2, J2), . . . , (Vk, Jk) be a complete system of eigenpairs ofX1, where

).

Vi = [vi1 . . . viji ], Ji =αi 1

. . .. . .. . . 1

αi

, i = 1, . . . , k,

andji is the order of the blockJi , so that

X1 = [V1 . . . Vk ]J1 0 0...

. . ....

0 0 Jk

[V1 . . . Vk ]−1.

For simplicity we suppose thatα1 is the eigenvalue having geometric multiplicity greater inC (and inP(λ) by Corollary 1) than inX1, and thatv11 is an eigenvector ofX1 corresponding toα1. Now, letvl1 bean eigenvector ofP(λ) corresponding toα1, that is not an eigenvector ofX1 (which exists by hypothesisSincev11 andvl1 are linearly independent, and then vectorsv11, . . . , v1j1, v21, . . . , v2j2, . . . , vk1, . . . , vkjkare linearly independent, there exists an infinite number of vectorswh = v1j1 + βvl1, for a nonzerocomplexβ, such that then vectorsv11, . . . , v1(j1−1),wh, v21, . . . , v2j2, . . . , vk1, . . . , vkjk are also linearlyindependent. Hence, there are infinitely many solvents

Yh = [V1(h) . . . Vk ]J1 0 0...

. . ....

0 0 Jk

[V1(h) . . . Vk ]−1,

where the columns ofV1(h) = [v11 . . . v1(j1−1) wh ] are also Jordan chains ofP(λ), of lengthj1,corresponding toα1, but are not Jordan chains ofX1 (corresponding toα1). ✷

We give an example to illustrate this situation.


P(X)=X2 +A1X+A2


A1 =[ −7 −2 −2

3/31 −203/31 8/31−13/31 −40/31 −231/31

], A2 =

[ 13 9 7−21/31 294/31 −36/3160/31 183/31 435/31

],

wherem= 2 andn= 3. A complete system of eigenpairs is

(V1, J1)=([v11 v12 ], J1

)=([−1 5

1 −3−2 1

],

[3 10 3

]),

(V2, J2)= (v21, J2)=([−2

1−1

],3

),

(V3, J3)=([v31 v32 v33 ], J3

)=([−2 −1 4

1 2 3−1 −1 2

],

[4 1 00 4 10 0 4

]).


The solvent [3 1 0]

r form

heg and

t

s

tionmatrix

X1 = [v11 v12 v31 ] 0 3 00 0 4

[v11 v12 v31 ]−1

=[−1 5 −2

1 −3 1−2 1 −1

][3 1 00 3 00 0 4

][−1 5 −21 −3 1

−2 1 −1

]−1

=[20/3 7 5/3

−2 −1 −17/3 5 13/3

],

has the eigenvalueα1 = 3 with geometric multiplicity 1, and inC the geometric multiplicity ofα1 is 2.Thus the conditions of Theorem 10 are satisfied. Now consider the vectors

wh = v12 + βv21,

whereβ is a nonzero complex number. Then there is an infinite number of solvents

Yh = [v11 wh v31 ][3 1 0

0 3 00 0 4

][v11 wh v31 ]−1,

for nonsingularTh = [v11 wh v31 ]. For example, withβ = 11,

Yh =[−1 −17 −2

1 8 1−2 −10 −1

][3 1 00 3 00 0 4

][−1 −17 −21 8 1

−2 −10 −1

]−1

=[−2/3 −15 −17/3

5/3 10 8/3−4/3 −6 2/3

],

is a solvent.

Reference [1] contains a study of infinite number of solvents in terms of the canonical triangulaof P(λ). WhenC has distinct eigenvalues, the maximum number of solvents is

(mn

n

), which occurs when

we have the Haar condition for themn eigenvectors ofP(λ) satisfied, as showed in Example 1. Tminimum number of solvents is obtained in the following theorem. The complete proof is very lonwill be omitted. Instead, we provide a brief sketch.

Theorem 11. LetP(X) be a matrix polynomial, andC the associated block companion matrix. IfC hasdistinct eigenvalues thenP(X) has at leastmn solvents.

Proof (Sketch). The proof is by induction onm. Consider the similarity matrixT , of order(m+ 1)n,of the block companion matrixD associated with a matrix polynomialQ(X) of degreem + 1. Thentake the submatrixS of T , of ordermn, on the top left-hand corner.S is nonsingular and by its forma(see Theorem 7) is a similarity matrix of a block companion matrixC of a matrix polynomialP(X) ofdegreem, and, by the induction hypothesis,P(X) has a minimum ofmn solvents. Making combinationof n linearly independent elements from the eigenvectors ofQ(λ) that are not eigenvectors ofP(λ) withthe eigenvectors ofP(λ), we get the remaining(m+ 1)n −mn solvents. ✷

4. Conclusions

We believe that the use of matrixS (Theorem 2) simplifies both the understanding and exemplificaof the process of construction of solvents. For instance, we can compare the Example 3 for a


polynomial with no solvents with the example in [11, p. 523]. From the numerical point of view, onlyan accurate study can show if the computation of the Jordan chains presents less complexity using the

as: ifnts ofne,t

1–845.

, Linear

matrix S instead of usingP(λ) directly. Our results on the number of solvents can be summarizeda matrixC is diagonalizable and its eigenvalues are distinct, then the minimum number of solvea P(X) is mn, and the maximum number is

(mn

n

). If a matrixC is diagonalizable and has at least o

multiple eigenvalue, then the number of solvents ofP(X) is infinite. If a matrixC is not diagonalizablethen the number of solvents ofP(X) can be zero, finite or infinite. It will be infinite ifP(X) has a solvenX1 which has an eigenvalue with geometric multiplicity greater inC than inX1.

Acknowledgements

We thank J. Vitória, T.P. de Lima and the Referees for their helpful suggestions.

References

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[10] P. Lancaster, P.N. Webber, Jordan chains for lambda matrices, Linear Algebra Appl. 1 (1968) 563–568.[11] P. Lancaster, M. Tismenetsky, The Theory of Matrices, 2nd Edition, Academic Press, New York, 1985.[12] C.C. MacDuffee, The Theory of Matrices, Chelsea, New York, 1946.[13] V.Y. Pan, Solving a polynomial equation: some history and recent progress, SIAM Rev. 39 (1997) 187–220.[14] W.E. Roth, A solution of the matric equationP(X)=A, Trans. Amer. Math. Soc. 30 (1928) 579–596.[15] W.E. Roth, On the unilateral equation in matrices, Trans. Amer. Math. Soc. 32 (1930) 61–80.

on solvents of matrix polynomials

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