An angle metric through the notion of Grassmann representative

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1 Electronic Journal of Linear Algebra Volume 18 Volume 18 (009 Article An angle metric through the notion of Grassmann representative Grigoris I. Kalogeropoulos Athanasios D. Karageorgos Athanasios A. Pantelous Follow this and additional works at: Recommended Citation Kalogeropoulos, Grigoris I.; Karageorgos, Athanasios D.; and Pantelous, Athanasios A.. (009, "An angle metric through the notion of Grassmann representative", Electronic Journal of Linear Algebra, Volume 18. DOI: This Article is brought to you for free and open access by Wyoming Scholars Repository. It has been accepted for inclusion in Electronic Journal of Linear Algebra by an authorized editor of Wyoming Scholars Repository. For more information, please contact

2 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 topologies 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 Plücker coordinates of the corresponding subspaces. 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, Plücker coordinates, Exterior algebra. AMS subject classifications. 15A75, 15A7, 3F45, 14M 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 Fx (t =Gx(t, or, in the discrete analogue, with autonomous singular (or regular linear difference systems Fx k+1 = Gx k, where x C n is the state control vector and F, G C m n (or F, 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, 008. Accepted for publication February 5, 009. Handling Editor: Michael J. Tsatsomeros. Department of Mathematics, University of Athens, Panepistimiopolis, GR Athens, Greece (gkaloger@math.uoa.gr, athkar@math.uoa.gr, apantelous@math.uoa.gr. 108

3 An Angle Metric Through the Notion of Grassmann Representative 109 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 L m,n = { L : L =(F, G;F, G C m n }, we define: a the flat matrix representation of L, [L] f = [ F G ] C m n,and [ ] F b the sharp matrix representation of L, [L] s = C m n. G Definition 1.. If L =(F, G L m,n,thenl is called non-degenerate when 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. If m = n, then non-degeneracy implies that L is a regular pair. By using these definitions, we can express the notion of strict equivalence as follows. Definition 1.5. Let L, L L m,n. a L and L are called strict equivalent (LE H L if and only if there exist non-singular matrices Q F n n and P F m m such that [ Q O [L ] f = P [L] f O Q ], or equivalently, [ P O [L ] s = O P ] [L] s Q.

4 110 G.I. Kalogeropoulos, A.D. Karageorgos, and A.A. Pantelous b L and L are called left strict equivalent (LE L H L if and only if there exists a non-singular matrix Q F n n such that [L ] f = P [L] f. Similarly, L and L are called right strict equivalent (LE R H L if and only if there exists a non-singular matrix Q F n n such that [L ] s =[L] s Q. For the pair L =(F, G L m,n, we can define four vector spaces as follows: X r f (L = rowspan F [L] f, X c f (L = colspan F [L] f, X r s (L = rowspan F [L] s, X c s (L = colspan F [L] s. It is clear that for every L, L L m,n with LEH LL,wehaveXf r (L =X f r (L and Xs r (L =X s r (L. Moreover, we observe that Xf r (L andx s r (L are under EH L-equivalence, but X f c (LandX s c (L are not always under EH L-equivalence. Also, for every L, L L m,n with LEH RL,weseethatXf c (L =X f c (L and Xs c (L =Xs c (L. Moreover, we observe that Xf c (L andx s c (L are under EH R-equivalence, but X f r (LandX s r (L are not always under ER H -equivalence. In the next lines, we provide a classification of the pairs considering their dimensions. Moreover, the 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 properties. Definition 1.6. For a non-degenerate pair L =(F, G L m,n,thenormalcharacteristic space, X N (L, is defined as follows: (i when m n, X N (L = Xf r (L, which is EL H -, (ii when m>n, X N (L = Xs c (L, which is EH R-. A straightforward consequence of (i is that for a regular pencil L =(F, G L n,n, we have that X N (L =X r f (L, and X r f (L =rowspan F [L] f = colspan F [L] t f X c s (L =colspan F [L] s = rowspan F [L] t s.

5 An Angle Metric Through the Notion of Grassmann Representative 111 Furthermore, in the case where the pair is non-degenerate and m n, we have rank [L] f = m, X N (L =X r f (L =rowspan F [L] f and dim X N (L =m. Therefore, we can identify X N (L with a point of the Grassmann manifold G ( m, F n. Note that if L 1 E L H L,thenX N (L 1 represents the same point on G ( m, F n as X N (L. Dimensions X r f (L X r s (L X c f (L X c s (L m n/ n/ <m n n m/ < m n/ <m n Regular, m = n E L H - E L H - X r f (L F n, E H - E L H - E L H - E L H - Xs r (L F m, E H - Xs r (L F n, E H - Xs r (L F n, E H - Xs r (L F n, E H - Xf c (L F m, E H - Xf c (L F m, E H - E R H - Xf c (L F m, E H - Xf c (L F m, E H - X c s (L F m, E H - E R H - E R H - E R H - E R H - Invariant spaces X r f (L, X r s (L. X r f (L, X c s (L X c f (L, X c s (L X r f (L, X c s (L X r f (L, X c s (L Now, let us consider the set of regular pairs. For a pair L =(F, G } L n,n, the equivalent class EH {L L = 1 L n,n :[L 1 ] f = P [L] f,p F n n, det P 0 can be considered as a representation of a point on G ( n, F n. This point is fully identified by X N (L 1, where L 1 EH L (L. Note that a point of G ( n, F n does not represent a pair, but only the left-equivalent class of pairs. Furthermore, we shall denote by Σ L 1 the set of all distinct generalized eigenvalues (finite and infinite of the pair L = (F, G L n,n, and by Σ L the set of normalized generalized eigenvectors (Jordan vectors included defined for every λ Σ L 1. We define Q = {( Σ L 1, ΣL : L Ln,n } and consider the function f : L n,n Q: L f (L = ( Σ L 1, ΣL Q. It is obvious that f {( 1 Σ L 1, } ΣL : L Ln,n = E L H (L, which means that f is EH L-. Thus, from the eigenvalue-eigenvector point of view, it is reasonable to identify the whole class EH L (L as a point of the Grassmann manifold. The study of topological properties of ordered regular pairs is intimately related with the generalized eigenvalue problem. The analysis above demonstrates that such properties may

6 11 G.I. Kalogeropoulos, A.D. Karageorgos, and A.A. Pantelous 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.. A New Angle Metric. In this session, we consider the Grassmann manifolds which are associated with elements of L m,n. In the case where m n, the manifold G ( m, F n is constructed, and on the other hand, when m n the manifolds G ( n, F m is the subject of investigation. The relationship between the two Grassmann manifolds and the pairs of L m,n is clarified by the following remark. Remark.1. (i For any V G ( m, F n,thereexistsaneh L -equivalence class from L m,n (m n, which is represented by a non-degenerate L =(F, G L m,n, such that V = rowspan [L] f. (ii For any V G ( n, F m,thereexistsaneh R-equivalence class from L m,n (m n, which is represented by a non-degenerate L =(F, G L m,n, such that V = colspan [L] s. Definition. ([8]. A real valued function d (, :G ( m, R n G ( m, R n R + o (m n is called an orthogonal metric if for every V 1, V, V 3 G ( m, R n (where V 1 = X N (L 1, V = X N (L, V 3 = X N (L 3, and L 1,L,L 3 L m,n are non-degenerate, it satisfies the following properties: (i d (X N (L 1, X N (L 0, and d (X N (L 1, X N (L = 0 if and only if L 1 EH LL, (ii d (X N (L 1, X N (L = d (X N (L, X N (L 1, (iii d (X N (L 1, X N (L d (X N (L 1, X N (L 3 + d (X N (L 3, X N (L, ( ] ] (iv d colspan [[L 1 ] tf U,colspan [[L ] tf U = d any orthogonal matrix U R m m. ( colspan [L 1 ] t f,colspan[l ] t f for Similarly, we can define the orthogonal metric on G ( n, R m.furthermore, we summarize some already known results for metrics on G ( m, R n, m n, and afterwards, we introduce the new angle metric. Theorem.3 ([1]. For any two points V L1, V L G ( m, R n such that V L1 = X r f (L 1=colspan R [L 1 ] t f and V L = X r f (L =colspan R [L ] t f, if we define X 1 = and ([L 1 ] tf 1 [L 1] f [L 1 ] t f and X = ([L ] tf 1 [L ] f [L ] t f,then J (V L1, V L = arccos [ det [ X 1 XX t X1 t ]] 1 (.1 are orthogonal metrics on G ( m, R n. d J (V L1, V L =sinj (V L1, V L (.

7 An Angle Metric Through the Notion of Grassmann Representative 113 Note that metrics (.1 and (. have been used extensively in [8]-[11], on the perturbation analysis of the generalized eigenvalue-eigenvector problem. These metrics have also been related to other metrics on the Grassmann manifold, for instance, the GAP between subspaces, sin θ (X, Y, etc.; see [8]-[11], [13]-[14]. Now, we denote L m,n /EH L as the quotient EL H-the collection of equivalence classes. It is easily derived that there is an one-to-one map ϕ : L m,n /E L H G ( m, R n such that E L H (L ϕ ( E L H (L = XN (L =X r f (L 1 = V L G ( m, R n. (.3 Considering map (.3, we can always associate an m-dimensional subspace V L of R n to a one-dimensional subspace of the vector space Λ ( m R n (see []-[5], or equivalently, a point on the Grassmann variety Ω (m, n of the projective space P v (R, ( n v = 1. This one-dimensional subspace of Λ ( m R n is described by decomposable multi-vectors called Grassmann representatives (or Plücker coordinates of m the corresponding subspace V L, which are denoted by g (V L. It is also known that if x,x Λ ( m R n are two Grassmann representatives of V L,thenx = λx where λ R\{0}. Definition.4. Let L = (F, G L m,n and V L G ( m, R n be a linear subspace of R n that corresponds to the equivalent class EH L (L. If x ( Λm R n is ( a Grassmann representative of V L and x =[x 0,x 1,...,x v ] t n, v = 1, then m we define the normal Grassmann representative of as the decomposable multi-vector which is given by x = sign(x v x x, if x v 0 sign(x i x x, if x v =0 and x i is the first non-zero component of x where sign (x i = { 1, if xi > 0 1, if x i < 0. Proposition.5. The normal Grassmann representative x of the subspace V L G ( m, R n is unique and x =1( denotes the Euclidean norm. Proof. Let x Λ m ( R n be another Grassmann representative of V L.Itisclear

8 114 G.I. Kalogeropoulos, A.D. Karageorgos, and A.A. Pantelous that x = λx for some λ R\{0}. By Definition.4, sign(x v x x = x, if x v = λx v 0 sign(x i x x, if x v = λx v =0 and x i = λx i = = x, λsign(λsign(x v λ x x, if x v 0 λsign(λsign(x i λ x x, if x v =0 and x i is the first non-zero component of x keeping in mind that λsign(λ λ =1. Thus, x is a unique Grassmann representative of the subspace V L. The property x = 1 is obvious. Definition.6 (The new angle metric. For V L1, V L the angle metric, (V L1, V L, by G ( m, R n, we define (V L1, V L = arccos x 1, x, (.4 where x 1, x are the normal Grassmann representatives of V L1, V L, respectively, and, denotes the inner product. In what it follows, we denote the set of strictly increasing sequences of l integers (1 l m chosenfrom1,,...,m; for example, Q,m = {(1,, (1, 3, (, 3,...}. ( m Clearly, the number of the sequences of Q l,m is equal to. l Furthermore, we assume that A =[a ij ] M m,n (R, where M m,n (R denotes the set of m n matrices over R. Letµ, ν be positive integers satisfying 1 µ m, 1 ν n, α =(i 1,...,i µ Q µ,m and β =(j 1,...j ν Q ν,n. Then A [α β ] M µ,ν (R denotes the submatrix of A formed by the rows i 1,i,...,i µ and the columns j 1,j,...,j ν. Finally, for 1 l min {m, n} the l-compound matrix (or l-adjugate ofa is ( ( m n the matrix whose entries are det {A [α β ]}, α Q l,m, β Q l,n, l l arranged lexicographically in α and β. ThismatrixisdenotedbyC l (A; see []-[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 Plücker coordinates of a space. Proposition.7. The angle metric (.4 is an orthogonal metric on the Grassmann manifold G ( m, R n. Proof. It is straightforward to verify conditions (i and (ii of Definition.. Condition (iii is derived by the corresponding inequality of angles between vectors

9 An Angle Metric Through the Notion of Grassmann Representative 115 which is a consequence of the -inequality on the unit sphere. Thus, we need to prove (iv. Let V Li = colspan R [L i ] t f, i =1,, and U be a m m non-singular matrix ( with U = λ. Then C m [L i ] t f U = C m ([L i ] t f U = λc m ([L i ] t f, i =1,, and consequently, ( ([L 1 ] tf U, [L ] tf U = [L 1 ] t f, [L ] t f. Moreover, if U is an orthogonal matrix, then an orthogonal metric is obtained. and This angle metric on G ( m, R n is related to the standard metrics (.1 and (.. Theorem.8. For every V L1, V L G ( m, R n, we have that (V L1, V L =J (V L1, V L sin { (V L1, V L } = d J (V 1, V =sin{j (V 1, V }. Proof. By using the Binet-Cauchy theorem [3], we obtain that det [X 1 X t X X t 1]=C m (X 1 C m (X t C m (X C m (X t 1 = C m (X 1 C t m (X C m (X C t m (X 1. By Theorem.3, we also have C m (X i = ] 1 [C m t [L i ] f C m (L i f Cm ([L t i ] f = g (V L i g (V Li = ε x t i, where ε = ±1, i =1,. Thus, since det [ X 1 X t X X1 t ] = x t 1 x xt x 1 = { x 1, x }, it follows that ( det [ X1 X t X X t 1] 1 = x 1,x. Then arccos ( det [ X 1 XX t X1 t ] 1 = arccos x 1, x J (V L1, V L = (V L1, V L, and sin { (V L1, V L } =sin{j (V L1, V L } = d J (V L1, V L.

10 116 G.I. Kalogeropoulos, A.D. Karageorgos, and A.A. Pantelous Remark.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. REFERENCES [1] C. Giannakopoulos. Frequency Assignment Problems of Linear Multivariable Problems: An Exterior Algebra and Algebraic Geometry Based Approach. PhD Thesis, City University, London, [] W. Greub. Multilinear Algebra, nd edition. Universitext, Springer-Verlag, New York- Heidelberg, [3] W.V.D. Hodge and D. Pedoe. Methods of Algebraic Geometry. Cambridge University Press, Boston, 195. [4] N. Karcanias and U. Başer. Exterior algebra and spaces of implicit systems: the Grassmann representative approach. Kybernetika (Prague, 30:1, [5] M. Marcus and H. Minc. A Survey of Matrix Theory and Matrix Inequalities. Allyn and Bacon, Boston, [6] G.W. Stewart. On the perturbation of pseudo-inverse, projections and linear least squares problems. SIAM Review, 19:634 66, [7] G.W. Stewart. Perturbation theory for rectangular matrix pencils. Linear Algebra and its Applications, 08/09:97 301, [8] G.W. Stewart and J.G. Sun. Matrix Perturbation Theory. Academic Press, Boston, [9] J.G. Sun. Invariant subspaces and generalized subspaces. I. Existence and uniqueness theorems. Mathematica Numerica Sinica (Chinese, (1:1 13, [10] J.G. Sun. Invariant subspaces and generalized subspaces. II. Perturbation theorems. Mathematica Numerica Sinica (Chinese, (:113 13, [11] J.G. Sun. Perturbation analysis for the generalized eigenvalue and the generalized singular value problem. Lecture Notes in Mathematics: Matrix Pencils, 973:1 44, [1] J.G. Sun. A note on Stewart s theorem for definite matrix pairs. Linear Algebra and its Applications, 48: , 198. [13] J.G. Sun. Perturbation of angles between linear subspaces. Journal of Computational Mathematics, 5:58 61, [14] Y. Zhang and L. Qiu. On the angular metrics between linear subspaces. Linear Algebra and its Applications, 41(1: , 007.