On the Lipschitz stability of (A,B)-invariant subspaces

Linear Algebra and its Applications 438 (2013) 182–190

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On the Lipschitz stability of (A, B)-invariant subspaces < Ferran Puerta a,∗ , Xavier Puerta b a b

Departament de Matemàtica Aplicada I, ETSEIB-UPC, Barcelona, Spain Institut d’Organització i Control, UPC, Spain

ARTICLE INFO

ABSTRACT

Article history: Received 28 January 2012 Accepted 17 July 2012 Available online 17 September 2012

Let T be the set of triples (A, B, S ) where S is an (A, B)-invariant subspace and let Fμ be the matrix having minimum norm such that (A + BFμ )(S ) ⊂ S. Then, if θ : T −→ Mm,n is the map defined by θ (A, B, S ) = Fμ and θ is continuous at (A, B, S ) a simple necessary and sufficient condition is given for the Lipschitz stability of S. It is shown that this continuity condition is satisfied in an open and dense subset of T and that the set of triples (A, B, S ) such that S is Lipschitz stable is also open and dense in T . © 2012 Elsevier Inc. All rights reserved.

Submitted by L. Rodman AMS classification: 93B10 15A60 Keywords: (A, B)-invariant subspace Lipschitz stability

1. Introduction Let A and B be n × n and n × m matrices, respectively, with coefficients in F, F being the field of real or complex numbers. (A, B)-invariant subspaces play an important role in the context of linear control systems, as shown in the basic references [1,5,13]. We recall the definition: a subspace S of Fn is said to be (A, B)-invariant, or controlled invariant in [1], if A(S ) ⊂ S + Im B. As it is pointed out in [9] the following definition of a stable invariant subspace is important in connection with computational aspects. A subspace S of Fn is said to be (A, B)-stable, or simply stable, if it is (A, B)-invariant and for every positive constant η, there exists a positive constant  such that for every pair (A , B ) satisfying

A − A + B − B < , < Partially supported by DGICYT MTM2007-67812-C02-02. ∗ Corresponding author. Tel.: +34 93 7520291; fax: +34 699096591. E-mail addresses: [email protected] (F. Puerta), [email protected] (X. Puerta). 0024-3795/$ - see front matter © 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.laa.2012.07.051

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F. Puerta, X. Puerta / Linear Algebra and its Applications 438 (2013) 182–190

183

there exists an (A , B )-invariant subspace S  such that

(S  , S )  η, where  denotes the gap metric between subspaces. A generalization of this concept was given in [5] (Theorem 15.8.1) : An (A, B)-invariant subspace S is said to be Lipschitz stable if there exist positive constants  and L such that for every pair (A , B ) satisfying (1) there exists an (A , B )-invariant subspace S  such that

(S  , S )  L(A − A  + B − B). It is clear that if S is Lipschitz stable, it is stable. Unlikely to stable and Lipschitz stable A-invariant subspaces, which are completely characterized in [5], there is not known a complete characterization of (A, B) stability and Lipschitz (A, B)-stability. Partial results can be found in [5,6,8,9,11]. In particular, in [5] it is shown that if the pair (A, B) is controllable, then every (A, B)-invariant subspace is Lipschitz stable. Our aim in this note is to give necessary and sufficient conditions for the Lipschitz stability when the following continuity condition is satisfied. Let T be the set of triples (A, B, S ) where S is a d-dimensional (A, B)-invariant subspace and denote by Fμ the m × n matrix having minimum euclidean norm (as a vector of Fnm ) such that (A + BFμ )(S ) ⊂ S. Then, if θ : T −→ Mm,n is the map defined by θ (A, B, S ) = Fμ and θ is continuous at (A, B, S ) a simple necessary and sufficient condition is given for the Lipschitz stability of S. Moreover, we show that this continuity condition is satisfied in an open and dense subset R of T and that the set of triples (A, B, S ) such that S is Lipschitz stable is also open and dense in T . In particular, this continuity condition is satisfied if Im B ∩ S = 0 . The above equality tells us that S is just a coasting subspace. Coasting subspaces are a wide class of (A, B)-invariant subspaces introduced by Willems [12] in his deep study of the structure of almost invariant subspaces. Hence, necessary and sufficient conditions for the Lipschitz stability of coasting subspaces are obtained. By duality this study can be transferred to the set of conditioned invariant subspaces and, in particular, to the set of tight subspaces. We recall that a subspace S is called tight if its orthogonal is coasting. This kind of subspaces has been extensively studied by Fuhrmann–Helmke (see, for example, [4]). Throughout this note, Mn,m denotes the set of n × m matrices with coefficients in F . As it is said F is the field of real or complex numbers. We write Mn,n = Mn and we denote by Grd (Fn ) the Grassmann manifold formed by the set of d-dimensional subspaces of Fn and by  the gap distance between two subspaces. We refer to [5] for its definition and properties. If A ∈ Mn,m the notation  A  stands, except in the proof of Theorem 4.1, for the usual operator norm induced by the Euclidean norms of Fn and Fm . If A is a matrix we denote by A∗ the transpose conjugate of A and by A† the Moore-Penrose inverse of A. As usual, if A is matrix we also denote by A the natural linear map associated to it. 2. Preliminaries Let A ∈ Mn , B ∈ Mn,m and let S be a d-dimensional subspace of Fn . We recall that S is said to be (A, B)-invariant or controlled invariant in [1] if there exists a matrix F ∈ Mm,n such that (A + BF )(S ) ⊂ S. The manifold N has been introduced in [8] as a key tool for obtaining sufficient conditions for the Lipschitz stability of an (A, B)-invariant subspace. We recall its definition. N

= {(F , A, B, S ) ∈ Mm,n × Mn × Mn,m × Grd (Fn ); (A + BF )(S ) ⊂ S }.

We will make use of the following results, proved in [8]. Proposition 2.1. N is a submanifold of Mm,n

× Mn × Mn,m × Grd (F)n of dimension n2 + 2nm.

Remark 2.2. Let Pd be the set of selfadjoint operators Pd

= {P ∈ Mn ; P = P ∗ , P 2 = P , rank P = d}.

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In [2] it is proved that Pd is a differentiable manifold and that the map Grd (Fn ) S

−→ Pd defined by

−→ X (X ∗ X )−1 X ∗ = P

with X ∈ Mn,d , Im X = S, is a diffeomorphism. We will say that P is the projection operator corresponding to S . Notice that Im P = S. Hence, the manifold N can be identified with the manifold

{(F , A , B , P ) ∈ Mm,n × Mn × Mn,m × Pd ; (In − P )(A + BF )P = 0}. It is clear that if we fix the matrices A and B, a subspace S is (A, B)-invariant if and only if the linear equation in F,

(In − P )BFP = −(In − P )AP , where P is the projection operator corresponding to S, has a solution. If vec stands for the usual vec function (see for example [7]), this equation is equivalent to

(P t ⊗ (In − P )B)vec(F ) = −vec((In − P )AP ).

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Hence, we have the following result, Lemma 2.3. With the previous notation, S only if rank(P t

= Im P is a d-dimensional (A, B)-invariant subspace if and

⊗ (In − P )B) = rank(P t ⊗ (In − P )B, vec((In − P )AP ).

Corollary 2.4. The set N(A,B,S ) of matrices F dim N(A,B,S )

∈ Mm,n such that (F , A, B, S ) ∈ N is a linear variety and

= nm − rank(P ⊗ (In − P )B)

where P is the projection operator corresponding to S. The following result allows to simplify the statements in Proposition 2.6, Proposition 3.2 and Theorem 4.1 (for a proof see [8] Lemma 4.3 or [6] Proposition 4.3) Lemma 2.5. Let S ∈ Grd (Fn ) be an (A, B)-invariant subspace and F ∈ Mm,n . Then, S is Lipschitz stable with respect to (A, B) if and only if it is Lipschitz stable with respect to (A + BF , B). So, while studying the Lipschitz stability of S ∈ Grd (Fn ), there is no loss of generality assuming that S is A-invariant. Then, taking a suitable basis of Fn the matrix representation of S and (A, B) takes de form ⎛⎛ ⎞ ⎛ ⎞⎞ ⎛ ⎞ Id A1 A2 ⎠ ⎝B1 ⎠⎠ ⎠ ⎝ ⎝ ⎝ S = Im (3) , (A, B) = , 0 A4 B2 0 with A1 a d × d matrix. Notice that then, P

=



Id 0 0 0

 .

Proposition 2.6. With the above notation the map α : Fn +2nm −→ N defined by ⎛⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎞      A B A 1 2 1      ⎠ , ⎝ ⎠ , F F , ⎝Id ⎠⎠ α(F1 , F2 , A1 , A2 , A4 , B1 , B2 , , Q ) = ⎝⎝ 1 2   B2 A3  A4 Q 2

with  A3 = Q  A4 Q + Q  A1 −  A2 Q neighborhood of (O, A, B, S ).

B2 F1 −  B2 F2 Q is a parametrization of N in a + Q B1 F1 + Q  B1 F2 Q − 

F. Puerta, X. Puerta / Linear Algebra and its Applications 438 (2013) 182–190

Proposition 2.7 ([8], Theorem 4.4). If the map (Q , F1 ) is equivalent, if rank(In−d

185

−→ QA1 − A4 Q − B2 F1 is surjective, or which

⊗ At1 − A4 ⊗ Id , (−B2 ) ⊗ Id ) = d(n − d)

then S is Lipschitz (A, B)-stable.

3. The set T Definition 3.1. We denote by T the set of triples (A, B, S ) with S a d-dimensional (A, B)-invariant subspace, that is, the set of triples (A, B, S ) such that (F , A, B, S ) ∈ N for some F ∈ Mm,n . If π : Mm,n × Mn × Mn,m × Grd (Fn ) −→ Mn × Mn,m × Grd (Fn ) is the natural projection , then T = π(N ). We endow T with the topology induced by the inclusion T Next proposition gives a local description of T .

⊂ Mn × Mn,m × Grd (Fn ). ⎞

⎛ Proposition 3.2. Let (A, B, S )

∈ T as in (3). Then , if  S = Im ⎝

⎞ ⎛ ⎞ ⎛ ⎞⎞ ⎛⎛   A  B I A ⎜⎜ 1 2 ⎟ ⎜ 1 ⎟ ⎜ d ⎟⎟ ⎠ , ⎝ ⎠ , ⎝ ⎠⎠ ⎝⎝   B2 A3  A4 Q

Id

⎠,

Q

∈T

in a neighborhood of (A, B, S ) if and only if rank(Q  B1

B2 ) = rank(Q  B2 , Q  A3 +  A4 Q ). − B1 −  A1 − Q  A2 Q + 

(4)

Proof. We take the coordinate of Nin a neighborhood of (A, B, S ) defined in Proposition 2.6.  system  I B, Im d ∈ T in a neighborhood of (A, B, S ) if and only if there Then, it can be checked that  A,  Q

exist F1

∈ Mm,d and F2 ∈ Mm,n−d such that

B1 F1 ) + ( B1 F2 )Q ) +  B2 F1 + ( B2 F2 )Q = 0. A1 +  A2 +  A3 +  A4 +  −Q ((

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If we write this equality as B2 )(F1 + F2 Q ) A3 +  A4 Q = (Q  −Q  A1 − Q  A2 Q +  B1 − 

(6)

the proposition follows.  Remark 3.3. B,  A,  S) (i) It is clear that (  A3

∈ T if and only if there is R ∈ Mm,d such that

B2 )R. A4 Q + (Q  = Q A1 + Q  A2 Q −  B1 − 

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So, we have an explicit way of obtaining any element of T in a neighborhood of (A, B, S ). Notice B,  A,  A3 . S ) can be chosen freely except those of  also that all the elements of ( (ii) The dimension of the linear variety N(A,B,S ) defined in Corollary 2.4 can also be obtained from Eq. (6). In fact, it can be checked that dim N(A,B,S )

B2 )). = nm − rank (Id ⊗ (Q  B1 − 

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If  P is the projection operator corresponding to  S, according to the mentioned lemma the following equality must hold ⎛ ⎞  B1 ⎠. B2 )) = rank(In −  P) ⎝ B1 −  (8) rank (Q   B2 This equality can be showed directly by taking into account that ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ Id ⎠ I ∗ ⎝ Id ⎠ −1 ∗  ⎝ P= ((Id Q ) ) (Id Q ) = ⎝ d ⎠ (Id + Q ∗ Q )−1 (Id Q ∗ ). Q Q Q In contrast to the set M of pairs (A, S ) where S is A-invariant (see [2]), T is not, in general, a differentiable manifold. Nevertheless, it contains an open and dense subset whose interest will become apparent in the next section. Proposition 3.4. Let R be the set of elements (A, B, S ) projection operator corresponding to S. Then, (i) R is a submanifold of Mn (ii) R is dense in T .

∈ T such that rank(In − P )B = m, where P is the

× Mn,m × Grd (Fn ) of dimension n2 + m(n + d) which is open in T .

Proof. Let (A, B, S ) ∈ T such that rank(In − P )B = m. Since for any F ∈ Mn,m , the map β : T −→ T defined by β(A, B, S ) = (A + BF , B, S ) is an homeomorphism, we can assume by virtue of Lemma 2.5, that the triple (A, B, S ) ∈ T is as in (3). Then, rank(In − P )B = m is equivalent to rank B2 = m and hence, there is an open neighborhood U(B,O) of (B, O) ∈ Mn,m × Mn−d,d , O being the null matrix   B1  1 − B  B, Q ) ∈ U(B,O) we have rank(Q B . It is of Mn−d,d , such that for every ( 2 ) = m, where B = clear now that the set W defined by B, Q ) ∈ Md × Md,n−d A1 ,  A2 ,  A3 ,  A4 ,  {(

 B2

A3 as in (7)} × Mn−d,d × Mn−d,n−d × U(B,O) ; 

(9)

is a subset of R which is open in T and that the map 1 , A 2 , A 3 , A 4 ,  B, Q , R) −→ (A B, Q ) (A1 , A2 , A4 ,  where  A3 is as in (7), defines a local coordinate system of R. This proves (i). Suppose now rank B2 < m. Then, in any neighborhood U of (B, O) ∈ Mn,m × Mn−d,d there is a B, Q ) ∈ U such that rank(Q  B2 ) = m . Hence, taking into account (8), it follows that R is B1 −  pair ( dense in T and (ii) is proved.  Remark 3.5. Notice that, since Im P (i) rank B = m, S (ii) rank(In − P )B

= ker(In − P ), the following two conditions are equivalent

∩ Im B = 0. = m.

Remark 3.6. From (i) of the previous Proposition we conclude that any element of R has a compact connected neighborhood as well as a contractible one. This fact will be needed for the application of Proposition 4.2. 4. The main result We fix (A, B, S ) ∈ T and we are interested in the unique Fμ ∈ N(A,B,S ) such that vec Fμ has minimum euclidean norm. Taking into account (2), we have (see for example [7]) vec Fμ

= (P t ⊗ (In − P )B)† vec((In − P )AP ).

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Then, the correspondence θ : T −→ Fnm defined by θ (A, B, S ) = vecFμ is a well defined map. As next theorem shows, the continuity of θ plays a key role in our aim of obtaining necessary and sufficient conditions for the Lipschitz stability. Thanks to Lemma 2.5 and the subsequent observations, we can assume ⎛ ⎞ ⎛ ⎞ A1 A2 Id ⎠ ⎝ ⎠. ⎝ , A= S = Im 0 0 A4 Throughout the proof of the next theorem, we consider the matrix norm corresponding to the inner product defined by < M , N >= trace MN ∗ , where M , N ∈ Mn,m . Theorem 4.1. With the above notation, suppose that the map θ Then, S is Lipschitz (A , B)-stable if and only if rank(In−d

: T −→ Fmn is continuous at (A, B, S ).

⊗ At1 − A4 ⊗ Id , (−B2 ) ⊗ Id ) = d(n − d).

(10)

Proof. We know, by Proposition 2.7, that (10) is a sufficient condition. In order to show that it is necessary, suppose now that it does not hold, but S is Lipschitz stable. Then, the map ψ : Mn−d,d × Mm,d −→ Mn−d,d defined by ψ(Q , F1 ) = QA1 − A4 Q − B2 F1 is not surjective. Hence, we can take A3 ∈ (Im ψ)⊥ , A3 = 0, the norm of A3 being as small as we want. Since S is Lipschitz stable, and taking into account that the operator norm and that corresponding to the inner product we are considering are equivalent norms (see, for example, [7] Theorem 10.4.1 ), we know that there exist two constants L and  such that if ⎛⎛ ⎞⎞ ⎛⎛ ⎞⎞ A1 A2 A1 A2 ⎝⎝ ⎠⎠ − ⎝⎝ ⎠⎠ = A3  <  0 A4 A3 A4 ⎞

⎛ there exists a matrix Q such that Im ⎝ matrix ⎝

Id

⎛⎛

⎠ is ⎝⎝

Q

⎞

⎛

Id

⎞ ⎛ A1 A2 A3 A4

⎠,⎝

⎞⎞ B1 B2

⎠⎠ - invariant and Q 

< LA3 , the

⎠ satisfying

Q ⎛⎛

⎞ ⎛ ⎞  A B A ⎜⎜ 1 2 ⎟ ⎜ 1 ⎟ ⎝⎝ ⎠+⎝ ⎠ F1 F2 A3 A4 B2



⎞⎛

⎞ ⎟ ⎝ Id ⎠ ⎠ Q

⎛

=⎝

⎞ Id

⎠R

Q

for some matrix R and where we take ⎛⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎞   A1 A2 B1 Id ⎝ ⎝ ⎠ ⎝ ⎠ , , Im ⎝ ⎠⎠ . F1 F2 = θ A3 A4 B2 Q A simple calculation shows that A3

= QA1 − A4 Q − B2 F1 + QA2 Q + QB1 F1 + QB1 F2 Q − B2 F2 Q

and since A3

∈ (Im ψ)⊥ , we have

 A3 2 =|< A3 , QA2 Q + QB1 F1 + QB1 F2 Q − B2 F2 Q >|  A3  Q  (α1  Q  +α2  F1  +α3  Q  F2  +α4  F2 )  α1 L2  A3 3 +L  A3 2 (α2  F1  +α4  F2 ) + α3 L2  A3 3  F2 

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F. Puerta, X. Puerta / Linear Algebra and its Applications 438 (2013) 182–190

Hence, if we denote  A3 1

= η one has

 α1 L2 η + L(α2  F1  +α4  F2 ) + α3 L2 η  F2  .

But the map θ is continuous at ⎛⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎞ A1 A2 B1 I ⎝⎝ ⎠ , ⎝ ⎠ , Im ⎝ d ⎠⎠ A3 A4 B2 O and at this point θ is equal to 0. So, for any A3 having small enough norm we can get α2  F1  +α4  F2 < δ . Since the constants η and δ can be taken arbitrarily small we have arrived to a contradiction. This shows that (10) is a necessary condition as claimed. 

In order to apply this theorem we have to analyze the continuity of the map θ . By definition,

θ (A, B, S ) = (P t ⊗ (In − P )B)† vec((In − P )AP ) P being the projection operator corresponding to S. It is clear that the map

(A, P ) −→ vec((In − P )AP ) is continuous. So, we are led to analyze the continuity of the map

(P , B) −→ (P t ⊗ (In − P )B)† . We will make use of the following proposition. Let T be a compact connected or a contractible topological space and denote by Co (T , Mkn,m ) the set of continuous maps A : T −→ Mn,m such that rank A(t ) = k for any t ∈ T. Then we have Proposition 4.2. Let A any t ∈ T .

∈ Co (T , Mkn,m ). Then there exists A† ∈ Co (T , Mkm,n ) such that A† (t ) = A(t )† for

For the proof we refer to [5] (Theorem 13.7.2) if T is compact connected or to [3] (IV-2-3) and the standard construction of the Moore-Penrose inverse, if T is contractible. Then, taking into account that the map

(P , B) −→ (P t ⊗ (In − P )B) is continuous, Remark 3.6 and the previous Proposition yields the following result Proposition 4.3. With the notation in Proposition 3.4 (see also Remark 3.5) the map θ is continuous at any triple (A, B, S ) ∈ R. Corollary 4.4. R is an open and dense subset of T such that to study the Lipschitz stability at any of its points, condition (10) can be applied. Remark 4.5. If (A, B, S ) ∈ R and F = C, in [6] a necessary and sufficient condition for the Lipschitz stability of S is given in terms of the spectra of certain matrices. Notice that according to the above proposition, for this kind of triples condition (10) reduces to a rank computation, the field F being R or C. So, in particular, condition (10) can be applied for studying Lipschitz stability of a given subspace when the pair (A, B) is perturbed. As it is said in the introduction, we have just showed that the continuity condition of θ is satisfied if S is a coasting subspace. In fact, condition (i) in (3.5) is not the definition given by Willems in [12], but

F. Puerta, X. Puerta / Linear Algebra and its Applications 438 (2013) 182–190

189

as it is known (see, for example, [10] Proposition 2.16) they are equivalent and it avoids introducing extra definitions not needed in this note. Hence we can state the following Theorem 4.6. If S is a coasting (A, B)-invariant subspace, then S is Lipschitz stable if and only if (10) holds true. Remark 4.7. By duality the above result can be transferred to conditioned invariant subspaces. In particular an analogous theorem can be stated for tight subspaces. Now, our goal is to study the subset Rls of R formed by the triples (A, B, S ) for which S is Lipschitz stable. For  we need to describe (10) in terms of a general triple (A, B, S ) ∈ T . Then, we have  this, S S = Im S12 , where , by reordering the basis of Fn if it is necessary, we may assume that S1 is a d × d     −1 S I invertible matrix. We take as a basis of S the columns of the matrix S12 S1 , so that S = Im Qd −1

with Q = S2 S1 . ¯ = A + BF, A¯ 1 = A1 + B1 F1 , A¯ 2 = A2 + B1 F2 Let F be such that (A + BF )S ⊂ S . Denote A ¯ 3 and A¯ 4 . Then we have the following proposition whose proof is left to the and similarly for A reader. ⎞

⎛ Proposition 4.8. If we take as a basis of F the columns of the matrix ⎝ n

B in this basis are, respectively, ⎛ ⎞ ⎛ ⎞ ˙ 1 A˙ 2 Id A ⎠ , B˙ ˙ =⎝ S˙ = ⎝ ⎠ , A ˙4 O 0 A

˙1 where A

⎛

=⎝

B˙ 1 B˙2

Id 0 Q Id

⎠, the matrices of S , A ¯ and

⎞ ⎠

= A¯ 1 + A¯ 2 Q , A˙ 2 = A¯ 2 , A˙ 4 = −Q A¯ 2 + A¯ 4 , B˙ 1 = B1 and B˙ 2 = B2 − QB1 .

It is clear that the map θ

(A˙ , B˙ , S ). Hence we have,

: T −→ Fnm is continuous at (A, B, S ) if and only if it is continuous at

Proposition 4.9. With the above notation, if only if rank(In−d

θ is continuous at (A, B, S ), S is Lipschitz stable if and

⊗ (A¯ 1 + A¯ 2 Q )t − (A¯ 4 − Q A¯ 2 ) ⊗ Id | (QB1 − B2 ) ⊗ Id ) = d(n − d).

(11)

As an application, we have Proposition 4.10. The set Rls is an open and dense subset in R. Proof. Let νi , i ∈ J be the set of all minors of order d(n − d) of the matrix in Eq. (11). Denote G the set of triples (A, B, S ) ∈ R such that νi = 0 for all i ∈ J. Taking into account that θ is continuous at R, we have from the previous proposition that if N = n2 + nm + d(n − d), Rls = (FN − G ) ∩ R = R − G. But it is clear that G is closed in FN , so that Rls is open, as claimed. In order to show that it is dense, we ¯ 1 + A¯ 2 Q )t − (A¯ 4 − Q A¯ 2 )⊗ Id ) = 0. consider the set H of elements (A, B, S ) ∈ R such that det(In−d ⊗(A It is clear that G ⊂ H; thus, to prove our claim it suffices to show that R − H is dense in R. To this end, take χ ∈ H and let V be a neighborhood of χ in R. We know that there is a neighborhood of χ in FN , W, such that V = R ∩ W . But, FN − H is dense in FN so that W ∩ (FN − H) = ∅ and we observe that the only elements of χ involved in H are A1 , A2 and Q . Hence (see Remark 3.3), W ∩ (FN − H) ∩ R = V ∩ (R − H) = ∅ and the proof is complete. 

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