Secondly, to establish a new angle metric which is described by decomposable multi-vectors called Grassmann representatives (or Pl¨ucker coordinates) of the corresponding sub- spaces

AN ANGLE METRIC THROUGH THE NOTION OF GRASSMANN REPRESENTATIVE^∗

GRIGORIS I. KALOGEROPOULOS^†, ATHANASIOS D. KARAGEORGOS^†, AND ATHANASIOS A. PANTELOUS^†

Abstract. The present paper has two main goals. Firstly, to introduce different metric topolo- gies on the pencils (F, G) associated with autonomous singular (or regular) linear differential or difference systems. Secondly, to establish a new angle metric which is described by decomposable multi-vectors called Grassmann representatives (or Pl¨ucker coordinates) of the corresponding sub- spaces. A unified framework is provided by connecting the new results to known ones, thus aiding in the deeper understanding of various structural aspects of matrix pencils in system theory.

Key words. Angle metric, Grassmann manifold, Grassmann representative, Pl¨ucker coordi- nates, Exterior algebra.

AMS subject classifications.15A75, 15A72, 32F45, 14M15.

1. Introduction - Metric Topologies on the Relevant Pencil (F, G). The perturbation analysis of rectangular (or square) matrix pencils is quite challenging due to the fact that arbitrarily small perturbations of a pencil can result in eigenvalues and eigenvectors vanishing. However, there are applications in which properties of a pencil ensure the existence of certain eigenpairs. Motivated by this situation, in this paper we consider the generalized eigenvalue problem (both for rectangular and for square constant coefficient matrices) associated with autonomous singular (or regular) linear differential systems

F x(t) =Gx(t),

or, in the discrete analogue, with autonomous singular (or regular) linear difference systems

F x_k+1=Gx_k,

wherex∈Cⁿ is the state control vector and F, G∈C^m×n (orF, G∈C^n×n). In the last few decades the perturbation analysis of such systems has gained importance, not only in the context of matrix theory, but also in numerical linear algebra and control theory; see [6], [8], [9], [11], [13].

∗Received by the editors November 5, 2008. Accepted for publication February 5, 2009. Handling Editor: Michael J. Tsatsomeros.

†Department of Mathematics, University of Athens, Panepistimiopolis, GR-15784 Athens, Greece ([email protected], [email protected], [email protected]).

108

The present paper is divided into two main parts. The first part deals with the topological aspects, and the second part with the “relativistic” aspects of matrix pencils. The topological results have a preliminary nature. By introducing different metric topologies on the space of pencils (F, G), a unified framework is provided aiding in the deeper understanding of perturbation aspects. In addition, our work connects the new angle metric (which in the authors’ view is more natural) to some already known results. In particular, the angle metric is connected to the results in [8]-[11] and the notion of deflating subspaces of Stewart [6], [8], and it is shown to be equivalent to the notion of e.d. subspace.

Some useful definitions and notations are required.

Definition 1.1. For a given pair L= (F, G)∈ Lm,n ∆

=

L:L= (F, G) ;F, G ∈C^m×n , we define:

a) theflat matrix representation ofL, [L]_f =

F −G

∈C^m×2n, and b) thesharp matrix representation ofL, [L]_s=

F

−G

∈C^2m×n.

Definition 1.2. IfL= (F, G)∈ Lm,n, thenLis callednon-degeneratewhen a) m≤n and rank [L]_f =m, or

b) m≥n and rank [L]_s=n.

Remark 1.3. In the case of the flat description, degeneracy implies zero row minimal indices, whereas, in the case of the sharp description, degeneracy implies zero column minimal indices.

Remark 1.4. Ifm=n, then non-degeneracy implies thatLis a regular pair.

By using these definitions, we can express the notion of strict equivalence as follows.

Definition 1.5. LetL, L∈ Lm,n.

a) LandLare calledstrict equivalent(LEHL) if and only if there exist non-singular matricesQ∈F^n×n andP ∈F^m×msuch that

[L]_f=P[L]_f

Q O O Q

,

or equivalently,

[L]_s=

P O O P

[L]_sQ.

b) L and L are called left strict equivalent (LE_H^LL) if and only if there exists a non-singular matrixQ∈F^n×n such that

[L]_f =P[L]_f.

Similarly,LandLare calledright strict equivalent(LE_H^RL) if and only if there exists a non-singular matrixQ∈F^n×n such that

[L]_s= [L]_sQ.

For the pairL= (F, G)∈ Lm,n, we can define four vector spaces as follows:

Xf^r(L)=^∆rowspan_F[L]_f, Xf^c(L)=^∆colspan_F[L]_f,

Xs^r(L)=^∆rowspan_F[L]_s, Xs^c(L)=^∆colspan_F[L]_s.

It is clear that for everyL, L ∈ Lm,nwithLE_H^LL, we haveXf^r(L) =Xf^r(L) and Xs^r(L) =Xs^r(L). Moreover, we observe thatXf^r(L) andXs^r(L) are invariant under E_H^L-equivalence, butX_f^c(L) andXs^c(L) are not always invariant underE_H^L-equivalence.

Also, for every L, L ∈ Lm,n with LE_H^RL, we see that Xf^c(L) = Xf^c(L) and Xs^c(L) =Xs^c(L). Moreover, we observe that Xf^c(L) andXs^c(L) are invariant under E_H^R-equivalence, butX_f^r(L) andXs^r(L) are not always invariant underE_H^R-equivalence.

In the next lines, we provide a classification of the pairs considering their di- mensions. Moreover, the invariant properties of the four vector subspaces are also investigated. The following classification is a simple consequence of the definitions and observations, so far. The table below demonstrates the type of subspace and the invariant properties.

Definition 1.6. For a non-degenerate pair L = (F, G) ∈ Lm,n, the normal- characteristic space, X_N(L), is defined as follows:

(i) whenm≤n,X_N(L)=^∆X_f^r(L), which isE_H^L-invariant, (ii) whenm > n,X_N(L)=^∆Xs^c(L), which isE_H^R-invariant.

A straightforward consequence of (i) is that for a regular pencilL= (F, G)∈ Ln,n, we have thatXN(L) =Xf^r(L),

Xf^r(L) =rowspan_F[L]_f =colspan_F[L]^t_f and

Xs^c(L) =colspan_F[L]_s=rowspan_F[L]^t_s.

Furthermore, in the case where the pair is non-degenerate and m ≤ n, we have rank[L]_f =m, XN(L) =X_f^r(L) =rowspan_F[L]_f and dimXN(L) =m. Therefore, we can identify XN(L) with a point of the Grassmann manifold G

m,F²ⁿ . Note that ifL₁E_H^LL, thenX_N(L₁) represents the same point onG

m,F²ⁿ

as X_N(L).

Dimensions Xf^r(L) Xs^r(L) Xf^c(L) Xs^c(L) Invariant spaces m≤n/2 E_H^L-

invariant E_H^L- invariant

X_f^c(L) ≡ F^2m, EH- invariant

Xs^c(L) ≡ F^2m, E_H- invariant

X_f^r(L), Xs^r(L).

n/2< m≤2n E_H^L- invariant

Xs^r(L) ≡ F^2m, E_H- invariant

X_f^c(L) ≡ F^2m, E_H- invariant

E_H^R- invariant

Xf^r(L), Xs^c(L)

n≤m/2<2m

Xf^r(L) ≡ F²ⁿ, E_H- invariant

Xs^r(L) ≡ Fⁿ, E_H- invariant

E_H^R- invariant

E_H^R- invariant

Xf^c(L), Xs^c(L)

n/2< m≤2n E_H^L- invariant

Xs^r(L) ≡ Fⁿ, E_H- invariant

Xf^c(L) ≡ F^m, E_H- invariant

E_H^R- invariant

Xf^r(L), Xs^c(L)

Regular,m=n E_H^L- invariant

Xs^r(L) ≡ Fⁿ, E_H- invariant

X_f^c(L) ≡ F^m, E_H- invariant

E_H^R- invariant

X_f^r(L), Xs^c(L)

Now, let us consider the set of regular pairs. For a pair L = (F, G) ∈ Ln,n, the equivalent class E_H^L = L₁∈ Ln,n: [L₁]_f =P[L]_f, P ∈F^n×n,detP = 0

can be considered as a representation of a point onG

n,F²ⁿ

. This point is fully identified byXN(L₁), whereL₁∈ E_H^L(L). Note that a point of G

n,F²ⁿ

does not represent a pair, but only the left-equivalent class of pairs. Furthermore, we shall denote by Σ^L₁ the set of all distinct generalized eigenvalues (finite and infinite) of the pair L= (F, G) ∈ Ln,n, and by Σ^L₂ the set of normalized generalized eigenvectors (Jordan vectors included) defined for everyλ∈Σ^L₁.

We defineQ=

Σ^L₁,Σ^L₂

:∀L∈ Ln,n

and consider the function f :Ln,n → Q:L→f(L)=^∆

Σ^L₁,Σ^L₂

∈ Q.

It is obvious thatf⁻¹

Σ^L₁,Σ^L₂

:∀L∈ Ln,n

=E_H^L(L), which means that f is E_H^L- invariant. Thus, from the eigenvalue-eigenvector point of view, it is reasonable to identify the whole classE_H^L(L) as a point of the Grassmann manifold. The study of topological properties of ordered regular pairs is intimately related with the general- ized eigenvalue problem. The analysis above demonstrates that such properties may

be studied on the appropriate Grassmann manifold. In the next section, a new angle metric is introduced from the point of view of metric topology on the Grassmann manifold.

2. A New Angle Metric. In this session, we consider the Grassmann man- ifolds which are associated with elements of Lm,n. In the case where m ≤ n, the manifold G

m,F²ⁿ

is constructed, and on the other hand, whenm ≥n the man- ifolds G

n,F^2m

is the subject of investigation. The relationship between the two Grassmann manifolds and the pairs ofLm,nis clarified by the following remark.

Remark 2.1. (i) For any V ∈G m,F²ⁿ

, there exists an E_H^L-equivalence class from Lm,n (m ≤ n), which is represented by a non-degenerate L= (F, G)∈ Lm,n, such thatV =rowspan[L]_f.

(ii) For anyV ∈G n,F^2m

, there exists anE_H^R-equivalence class fromLm,n(m≥n), which is represented by a non-degenerateL= (F, G)∈ Lm,n, such that V=colspan [L]_s.

Definition 2.2 ([8]). A real valued functiond(·,·) :G

m,R²ⁿ

×G

m,R²ⁿ

→ R⁺o (m ≤ n ) is called an orthogonal invariant metric if for every V1,V2,V3 ∈ G

m,R²ⁿ

(where V₁ = X_N(L₁), V₂ =X_N(L₂), V₃ = X_N(L₃), and L₁, L₂, L₃ ∈ Lm,n are non-degenerate), it satisfies the following properties:

(i) d(XN(L₁),XN(L₂))≥0, andd(XN(L₁),XN(L₂)) = 0 if and only ifL₁E_H^LL₂, (ii) d(XN(L₁),XN(L₂)) =d(XN(L₂),XN(L₁)),

(iii) d(X_N(L₁),X_N(L₂))≤d(X_N(L₁),X_N(L₃)) +d(X_N(L₃),X_N(L₂)), (iv) d

colspan

[L₁]^t_fU

, colspan

[L₂]^t_fU

= d

colspan[L₁]^t_f, colspan[L₂]^t_f

for any orthogonal matrixU ∈R^m×m.

Similarly, we can define the orthogonal invariant metric onG

n,R^2m

. Further- more, we summarize some already known results for metrics onG

m,R²ⁿ

,m≤n, and afterwards, we introduce the new angle metric.

Theorem 2.3 ([12]). For any two points VL₁,VL₂∈G

m,R²ⁿ

such that VL₁ =Xf^r(L₁) =colspan_R[L₁]^t_f and VL₂ =Xf^r(L₂) =colspan_R[L₂]^t_f, if we defineX₁=

[L₁]^t_f[L₁]_{f −}¹₂

[L₁]^t_f andX₂=

[L₂]^t_f[L₂]_{f −}¹₂

[L₂]^t_f, then J(VL₁,VL₂) = arccos

det

X₁X₂^tX₂X₁^t1₂

(2.1) and

d_J (VL₁,VL₂) = sinJ(VL₁,VL₂) (2.2) are orthogonal invariant metrics on G

m,R²ⁿ .

Note that metrics (2.1) and (2.2) have been used extensively in [8]-[11], on the perturbation analysis of the generalized eigenvalue-eigenvector problem. These met- rics have also been related to other metrics on the Grassmann manifold, for instance, the “GAP” between subspaces, sinθ(X, Y), etc.; see [8]-[11], [13]-[14].

Now, we denoteLm,n/E_H^Las the quotientE_H^L-the collection of equivalence classes.

It is easily derived that there is an one-to-one map ϕ:Lm,n/E_H^L →G

m,R²ⁿ

such that

E_H^L(L)→ϕ

E_H^L(L)_∆

=XN(L) =Xf^r(L₁)=^∆VL∈G

m,R²ⁿ

. (2.3)

Considering map (2.3), we can always associate anm-dimensional subspaceVLofR²ⁿ to a one-dimensional subspace of the vector space Λ^m

R²ⁿ

(see [2]-[5]), or equiv- alently, a point on the Grassmann variety Ω (m,2n) of the projective spaceP^v(R), v=

2n m

−1. This one-dimensional subspace of Λ^m R²ⁿ

is described by decom- posable multi-vectors called Grassmann representatives (or Pl¨ucker coordinates) of the corresponding subspaceVL, which are denoted by g(VL). It is also known that if x, x ∈ Λ^m

R²ⁿ

are two Grassmann representatives of VL, then x = λx where λ∈R\ {0}.

Definition 2.4. Let L = (F, G) ∈ Lm,n and VL ∈ G

m,R²ⁿ

be a linear subspace ofR²ⁿ that corresponds to the equivalent class E_H^L(L). If x∈Λ^m

R²ⁿ is a Grassmann representative ofVL and x= [x₀, x₁, . . . , xv]^t, v =

2n m

−1, then we define the normal Grassmann representative of as the decomposable multi-vector which is given by

x=





sign(x_v)

x₂ x, if xv= 0

sign(x_i)

x₂ x, if xv= 0 and xi is the first non-zero component of x

wheresign(xi) =

1, if xi >0

−1, if xi <0 .

Proposition 2.5. The normal Grassmann representativexof the subspaceVL∈ G

m,R²ⁿ

is unique andx₂= 1 ( · ₂ denotes the Euclidean norm).

Proof. Letx ∈Λ^m R²ⁿ

be another Grassmann representative ofVL. It is clear

thatx=λxfor some λ∈R\ {0}. By Definition 2.4, x =







sign(^x_v)

x₂ x, if x_v=λxv = 0

sign(^xi)

x₂ x, if x_v =λxv= 0 and x_i =λxi

=





λsign(λ)sign(x_v)

|λ|x₂ x, if xv = 0

λsign(λ)sign(x_i)

|λ|x₂ x, if xv= 0 and xi is the first non-zero component ofx

=x ,

keeping in mind that ^λsign_|λ|^(λ) = 1. Thus,xis a unique Grassmann representative of the subspaceVL. The propertyx₂= 1 is obvious.

Definition 2.6 (The new angle metric). ForVL₁,VL₂ ∈G

m,R²ⁿ

, we define the angle metric,(VL₁,VL₂), by

(VL₁,VL₂)= arccos^∆ |x₁,x₂|, (2.4) wherex₁,x₂are the normal Grassmann representatives ofVL₁,VL₂, respectively, and

·,·denotes the inner product.

In what it follows, we denote the set of strictly increasing sequences oflintegers (1≤l ≤m) chosen from 1,2, . . . , m; for example, Q₂,m ={(1,2),(1,3),(2,3), . . .}.

Clearly, the number of the sequences ofQl,mis equal to m

l

.

Furthermore, we assume that A = [aij] ∈ Mm,n(R), whereMm,n(R) denotes the set ofm×nmatrices overR. Letµ, ν be positive integers satisfying 1≤µ≤m, 1 ≤ ν ≤ n, α = (i₁, . . . , iµ) ∈ Qµ,m and β = (j₁, . . . jν) ∈ Qν,n. Then A[α|β] ∈ Mµ,ν(R) denotes the submatrix ofAformed by the rowsi₁, i₂, . . . , iµand the columns j₁, j₂, . . . , jν.

Finally, for 1 ≤ l ≤min{m, n} the l-compound matrix (or l-adjugate) of A is the

m l

× n

l

matrix whose entries are det{A[α|β]}, α ∈ Ql,m, β ∈ Ql,n, arranged lexicographically inαandβ. This matrix is denoted byCl(A); see [2]-[5] for more details about this interesting matrix. Moreover, it should be stressed out that a compound matrix is in fact a vector corresponding to the Grassmann representative (or Pl¨ucker coordinates) of a space.

Proposition 2.7. The angle metric (2.4) is an orthogonal invariant metric on the Grassmann manifold G

m,R²ⁿ .

Proof. It is straightforward to verify conditions (i) and (ii) of Definition 2.2.

Condition (iii) is derived by the corresponding inequality of angles between vectors

which is a consequence of the ∆-inequality on the unit sphere. Thus, we need to prove (iv). LetVL_i =colspan_R[Li]^t_f,i= 1,2, andU be am×mnon-singular matrix with |U| =λ. Then Cm

[Li]^t_fU

=Cm

[Li]^t_f

|U| = λCm

[Li]^t_f

, i = 1,2, and consequently,

[L₁]^t_fU,[L₂]^t_fU

=

[L₁]^t_f,[L₂]^t_f

.

Moreover, if U is an orthogonal matrix, then an orthogonal invariant metric is ob- tained.

This angle metric onG

m,R²ⁿ

is related to the standard metrics (2.1) and (2.2).

Theorem 2.8. For every VL₁,VL₂∈G

m,R²ⁿ

, we have that (VL₁,VL₂) =J (VL₁,VL₂)

and

sin{(VL₁,VL₂)}=d_J(V₁,V₂) = sin{J(V₁,V₂)}.

Proof. By using the Binet-Cauchy theorem [3], we obtain that det [X₁X₂^tX₂X₁^t] =Cm(X₁)Cm(X₂^t)Cm(X₂)Cm(X₁^t)

=Cm(X₁)C_m^t (X₂)Cm(X₂)C_m^t (X₁). By Theorem 2.3, we also have

Cm(Xi) =

C_m^t [Li]_fCm(Li)_{f −}¹₂

C_m^t

[Li]_f

= g(VL_i) g(VL_i)

2

=εx^t_i, whereε=±1,i= 1,2.

Thus, since

det

X₁X₂^tX₂X₁^t

=x^t₁x₂x^t₂x₁={x₁,x₂}², it follows that

det

X₁X₂^tX₂X₁^t1₂

=|x₁, x₂|. Then

arccos det

X₁X₂^tX₂X₁^t1₂

= arccos|x₁,x₂| ⇔ J(VL₁,VL₂) =(VL₁,VL₂), and

sin{(VL₁,VL₂)}= sin{J(VL₁,VL₂)}=d_J(VL₁,VL₂).

Remark 2.9. When m =n, this angle metric is suitable for the perturbation analysis for the generalized eigenvalue-eigenvector problem.

Acknowledgments. The authors are grateful to the anonymous referees for their insightful comments and suggestions.

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