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    Controllability of continuous bimodal linear systems  Open access

     Ferrer Llop, Jose; Pacha Andujar, Juan Ramon; Peña Carrera, Marta
    Mathematical problems in engineering
    Date of publication: 2013-05-01
    Journal article

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    We consider bimodal linear systems consisting of two linear dynamics acting on each side of a given hyperplane, assuming continuity along the separating hyperplane. We prove that the study of controllability can be reduced to the unobservable case, and for these ones we obtain a simple explicit characterization of controllability for dimensions 2 and 3, as well as some partial criteria for higher dimensions

    We consider bimodal linear systems consisting of two linear dynamics acting on each side of a given hyperplane, assuming continuity along the separating hyperplane. We prove that the study of controllability can be reduced to the unobservable case, and for these ones we obtain a simple explicit characterization of controllability for dimensions 2 and 3, as well as some partial criteria for higher dimensions.

  • Determinant of a matrix that commutes with a Jordan matrix

     Montoro López, Maria Eulalia; Ferrer Llop, Jose; Mingueza, David
    Linear algebra and its applications
    Date of publication: 2013-10-30
    Journal article

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    Let F be an arbitrary field, Mn(F) the set of all matrices n×n over F and J¿Mn(F) a Jordan matrix. In this paper, we obtain an explicit formula for the determinant of any matrix that commutes with J, i.e., the determinant of any element T¿Z(J), the centralizer of J. Our result can also be extended to any T'¿Z(A), where A¿Mn(F), can be reduced to J=S-1AS. This is because T=S-1T'S¿Z(J), and clearly View the MathML source. If F is algebraically closed, any matrix A can be reduced in this way to a suitable J. In order to achieve our main result, we use an alternative canonical form W¿Mn(F) called the Weyr canonical form. This canonical form has the advantage that all matrices K¿Z(W) are upper block triangular. The permutation similarity of T¿Z(J) and K¿Z(W) is exploited to obtain a formula for the determinant of T. The paper is organized as follows: Section 2 contains some definitions and notations that will be used through all the paper. In Section 3, matrices T¿Z(J) are described and the determinant of T is computed in a particular case. In Section 4, we recall the Weyr canonical form W of a matrix and the corresponding centralizer Z(W). A formula to compute the determinant of any K¿Z(W) is rewritten. Finally, in Section 5 an explicit formula for the determinant of any T¿Z(J) is obtained.

  • Structural stability of planar bimodal linear systems

     Ferrer Llop, Jose; Peña Carrera, Marta; Susin Sanchez, Antonio
    International Conference on Numerical Analysis and Applied Mathematics
    Presentation's date: 2013-09
    Presentation of work at congresses

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    We consider bimodal linear dynamical systems consisting of two linear dynamics acting on each side of a given hyperplane, assuming continuity along the separating hyperplane. Focusing in the planar case, we describe which of these systems are structurally stable

  • Differentiable families of stabilizers for planar bimodal linear control systems

     Ferrer Llop, Jose; Magret Planas, Maria Dels Dolors; Peña Carrera, Marta
    International Conference on Numerical Analysis and Applied Mathematics
    Presentation's date: 2013-09
    Presentation of work at congresses

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    We consider bimodal linear control systems consisting of two subsystems acting on each side of a given hyperplane, assuming continuity along the separating hyperplane. For a differentiable family of controllable planar ones, we construct a differentiable family of feedbacks which point wise stabilizes both subsystems

  • Perturbations preserving conditioned invariant subspaces

     Compta Creus, Albert; Ferrer Llop, Jose; Peña Carrera, Marta
    Mathematical methods in the applied sciences
    Date of publication: 2012-02
    Journal article

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    Given the set of matrix pairsM Mm,n.C/ Mn.C/ keeping a subspace S Cn invariant, we obtain a miniversal deformation of a pair belonging to an open dense subset ofM. It generalizes the known results when S is a supplementary subspace of the unobservable one.

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    Learning automation to teach mathematics  Open access

     Ferrer Llop, Jose; Peña Carrera, Marta; Ortiz Caraballo, Carmen
    Date of publication: 2012-07
    Book chapter

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    Classification of monogenic invariant subspaces and uniparametric linear control systems  Open access

     Compta Creus, Albert; Ferrer Llop, Jose
    Date: 2012-12
    Report

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    The classification of the invariant subspaces of an endomorphism has been an open problem for a long time, and it is a ”wild” problem in the general case. Here we obtain a full classification for the monogenic ones. Some applications are derived: in particular, canonical forms for uniparametric linear control systems, non necessarily controllable, with regard to linear changes of state variables

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    Miniversal deformations of observable marked matrices  Open access

     Compta Creus, Albert; Ferrer Llop, Jose; Peña Carrera, Marta
    Date: 2012
    Report

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    Given the set of vertical pairs of matrices M¿ Mm,n(C)×Mn(C) keeping the subspace Cd×{0} ¿ Cn invariant, we compute miniversal deformations of a given pair when it is observable and the subspace Cd × {0} is marked. Moreover, we obtain the dimension of the orbit, characterize the structurally stable vertical pairs and study the effect of each deformation parameter.

  • Estructuras geométricas de los sistemas de control lineales, lineales a trozos y sistemas conmutados

     Compta Creus, Albert; Clotet Juan, Jose; Garcia Planas, Maria Isabel; Magret Planas, Maria Dels Dolors; Peña Carrera, Marta; Puerta Coll, Francisco Javier; Montoro López, Maria Eulalia; Puerta Sales, Fernando; Ferrer Llop, Jose
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    Geometric structure of single/combined equivalence classes of a controllable pair  Open access

     Compta Creus, Albert; Ferrer Llop, Jose; Peña Carrera, Marta
    Electronic journal of linear algebra
    Date of publication: 2011-11
    Journal article

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    Given a pair of matrices representing a controllable linear system, its equivalence classes by the single or combined action of feedbacks, change of state and input variables, as well as their intersection are studied. In particular, it is proved that they are differentiable manifolds and their dimensions are computed. Some remarks concerning the effect of different kinds of feedbacks are derived.

  • On the effect of friend feedbacks

     Compta Creus, Albert; Ferrer Llop, Jose; Peña Carrera, Marta
    Mathematical methods in the applied sciences
    Date of publication: 2011
    Journal article

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    Given S an (A,B)-invariant subspace, we prove that the set of friend feedbacks is a linear variety, which can be considered as the direct sum of the feedbacks of the restriction to S and the co-restriction to S ⊥. In particular, when the natural controllability hypothesis hold, both pole assignments are simultaneously possible bymeans of a convenient friend feedback. Copyright

  • Estructuras geométricas de los sistemas lineales de control y su extensión a los sistemas conmutados

     Clotet Juan, Jose; Montoro López, Maria Eulalia; Peña Carrera, Marta; Compta Creus, Albert; Puerta Coll, Francisco Javier; Puerta Sales, Fernando; Garcia Planas, Maria Isabel; Magret Planas, Maria Dels Dolors; Ferrer Llop, Jose
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    Perturbations preserving conditioned invariant subspaces  Open access

     Peña Carrera, Marta; Compta Creus, Albert; Ferrer Llop, Jose
    International Conference on Circuits, Systems, Signals
    Presentation's date: 2010-09
    Presentation of work at congresses

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    Given the set of matrix pairs M ⊂ Mm,n(C) × Mn(C) keeping a subspace S ⊂ Cn invariant, we obtain a miniversal deformation of a pair belonging to an open dense subset of M. It generalizes the known results when S is a supplementary subspace of the unobservable one.

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    Miniversal Deformations of Bimodal Picewise Linear Systems  Open access

     Ferrer Llop, Jose; Magret Planas, Maria Dels Dolors; Pacha Andujar, Juan Ramon; Peña Carrera, Marta
    Meeting on Linear Algebra Matrix Analysis and Applications
    Presentation's date: 2010-06
    Presentation of work at congresses

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    Bimodal linear systems are those consisting of two linear systems on each side of a given hyperplane, having continuous dynamics along that hyperplane. In this work, we focus on the derivation of (orthogonal) miniversal deformations, by using reduced forms.

    Keywords: Bimodal piecewise linear system, miniversal deformations, reduced forms.

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    Distance from a controllable switched linear system to an uncontrollable one  Open access

     Clotet Juan, Jose; Ferrer Llop, Jose; Magret Planas, Maria Dels Dolors
    Meeting on Linear Algebra Matrix Analysis and Applications
    Presentation's date: 2010-06-02
    Presentation of work at congresses

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    The set of controllable switched linear systems is an open set in the space of all switched linear systems. Then it makes sense to compute the distance from a controllable switched linear system to the set of uncontrollable systems. In this work we obtain an upper bound for such distance.

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    Learning engineering to teach mathematics  Open access

     Ferrer Llop, Jose; Peña Carrera, Marta; Ortiz, Carmen
    Annual Frontiers in Education Conference
    Presentation's date: 2010-10-27
    Presentation of work at congresses

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    The Bologna process is a good opportunity to bring together first-year mathematics courses of engineering degrees and technology courses offered in subsequent years. In fact, the Faculty Council has decided that 20% of the credits from basic courses must be related to technological applications. To this end, during the past academic year a mathematical engineering seminar was held with each session dealing with one technological discipline. The main goal of the seminar, which relied on the presence of speakers from both mathematics and engineering departments, was to identify the most commonly used mathematical tools. Furthermore, a set of exercises and some guidelines addressed to faculty lacking an engineering background were created. Here, we present some of this material: first, a summary of the collection of exercises illustrating the use of Linear Algebra in different engineering areas such as Mechanical Engineering, Control and Automation, and second, some exercises

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    Geometric structure of the equivalence classes of a controllable pair  Open access

     Peña Carrera, Marta; Compta Creus, Albert; Ferrer Llop, Jose
    International Conference on Circuits, Systems, Signals
    Presentation's date: 2010-09
    Presentation of work at congresses

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    Given a pair of matrices representing a controllable linear system, we study its equivalence classes by the single or combined action of feedbacks and change of state and input variables, as well as their intersections. In particular, we prove that they are differentiable manifolds and we compute their dimensions. Some remarks concerning the effect of different kinds of feedbacks are derived.

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    Local differentiable pole assignment  Open access

     Compta Creus, Albert; Ferrer Llop, Jose; Peña Carrera, Marta
    Linear and multilinear algebra
    Date of publication: 2010
    Journal article

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    Given a general local differentiable family of pairs of matrices, we obtain a local differentiable family of feedbacks solving the pole assignment problem, that is to say, shifting the spectrum into a prefixed one. We point out that no additional hypothesis is needed. In fact, simple approaches work in particular cases (controllable pairs, constancy of the dimension of the controllable subspace, and so on). Here the general case is proved by means of Arnold’s techniques: the key point is to reduce the construction to a versal deformation of the central pair; in fact to a quite singular miniversal one for which the family of feedbacks can be explicitly constructed. As a direct application, a differentiable family of stabilizing feedbacks is obtained, provided that the central pair is stabilizable.

  • Learning engineering to teach mathematics

     Ferrer Llop, Jose; Peña Carrera, Marta; Ortiz Caraballo, Carmen
    Frontiers in education conference
    Date of publication: 2010-10
    Journal article

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  • Planar bimodal piecewise linear systems. Bifurcation diagrams

     Ferrer Llop, Jose; Magret Planas, Maria Dels Dolors; Pacha Andujar, Juan Ramon; Peña Carrera, Marta
    Boletín de la Sociedad Española de Matemática Aplicada
    Date of publication: 2010
    Journal article

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  • Bimodal piecewise linear dynamical systems. Reduced forms

     Ferrer Llop, Jose; Magret Planas, Maria Dels Dolors; Peña Carrera, Marta
    International journal of bifurcation and chaos
    Date of publication: 2010
    Journal article

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    Output regulation problem for differentiable families of linear systems  Open access

     Compta Creus, Albert; Ferrer Llop, Jose; Peña Carrera, Marta
    Mathematical problems in engineering
    Date of publication: 2010
    Journal article

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    Given a family of linear systems depending on a parameter varying in a differentiable manifold, we obtain sufficient conditions for the existence of a (global or local) differentiable family of controllers solving the output regulation problem for the given family. Moreover, we construct it when these conditions hold.

  • Use of reduced forms in the disturbance decoupling problem

     Compta Creus, Albert; Ferrer Llop, Jose; Peña Carrera, Marta
    Linear algebra and its applications
    Date of publication: 2009-03
    Journal article

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    Specific algorithms, such as those involving the supremal of the invariant subspaces contained in a suitable subspace, are known to be able to test whether a disturbance decoupling problem (DDP) is solvable. Here, by reducing the system to its Molinari form, we obtain an alternative description of this supremal object and compute its dimension. Hence we have a general result for solving the decoupling provided that a Molinari basis is known. In particular, a necessary numerical condition for it is derived. The same technique is applied to the DDPS, that is, when stability of the decoupled closed loop system is required.

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    Output regulation problem for differentiable families of linear systems  Open access

     Compta Creus, Albert; Ferrer Llop, Jose; Peña Carrera, Marta
    AIP Conference proceedings
    Date of publication: 2009
    Journal article

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    Given a family of linear systems depending on a parameter varying in a differentiable manifold, we obtain sufficient conditions for the existence of a (global or local) differentiable family of controllers solving the output regulation problem for the given family. Moreover, we construct it when these conditions hold.

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    On the effect of friend feedbacks  Open access

     Compta Creus, Albert; Ferrer Llop, Jose; Peña Carrera, Marta
    AIP Conference proceedings
    Date of publication: 2009
    Journal article

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    Given S an (A;B)-invariant subspace, we prove that the set of friend feedbacks is a (nm¡md+dq)-dimensional linear variety, which can be considered as the direct sum of the feedbacks of the restriction to S and the co-restriction to S?. In particular, if (A;B) is controllable and S is a controllability subspace, both pole assignments are simultaneously possible by means of a convenient friend feedback.

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    Switched singular linear systems  Open access

     Clotet Juan, Jose; Ferrer Llop, Jose; Magret Planas, Maria Dels Dolors
    Mediterranean Conference on Control and Automation
    Presentation's date: 2009-06
    Presentation of work at congresses

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    We consider switched singular linear systems and determine the set of reachable/controllable states. We derive necessary and sufficient conditions for such a system to be reachable/controllable when an “equisingularity condition” holds.

  • Use of reduced forms in the disturbance decoupling problem

     Compta Creus, Albert; Ferrer Llop, Jose; Peña Carrera, Marta
    Date: 2008-04
    Report

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  • Realización de perturbaciones

     Compta Creus, Albert; Ferrer Llop, Jose; Peña Carrera, Marta
    Séptimo Encuentro Plenario de Teoría de Sistemas
    Presentation's date: 2007-01-01
    Presentation of work at congresses

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  • Geometría de las perturbaciones de sistemas lineales de control

     Puerta Coll, Francisco Javier; Compta Creus, Albert; Magret Planas, Maria Dels Dolors; Peña Carrera, Marta; Clotet Juan, Jose; Ferrer Llop, Jose; Puerta Sales, Fernando
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  • Local differentiable pole assignment

     Compta Creus, Albert; Ferrer Llop, Jose; Peña Carrera, Marta
    Date: 2007-02
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  • Geometric structure of the equivalence classes of a controllable pair

     Compta Creus, Albert; Ferrer Llop, Jose; Peña Carrera, Marta
    Date: 2007-05
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  • Estructura diferenciable de las clases de equivalencia de un par controlable

     Compta Creus, Albert; Ferrer Llop, Jose; Peña Carrera, Marta
    Congreso de Ecuaciones Diferenciales y Aplicaciones / Congreso de Matemática Aplicada
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  • Uso de formas reducidas en el "disturbance decoupling problem"

     Ferrer Llop, Jose
    Séptimo Encuentro Plenario de Teoría de Sistemas
    Presentation's date: 2007-01-01
    Presentation of work at congresses

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  • Estructura geométrica de las clases de una pareja controlable

     Ferrer Llop, Jose
    Vigésimo quinto aniversario del grupo de álgebra lineal de la Univ. del País Vasco
    Presentation's date: 2006-09-21
    Presentation of work at congresses

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  • Use of reduced forms in the Disturbance Decoupling Problem

     Compta Creus, Albert; Ferrer Llop, Jose; Peña Carrera, Marta
    Date: 2006-12
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  • Deformacions d'operadors i subespais associats a sistemes lineals multivariables

     Peña Carrera, Marta
    Defense's date: 2005-03-11
    Department of Applied Mathematics I, Universitat Politècnica de Catalunya
    Theses

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  • Dimension of the orbit of marked subspaces

     Compta Creus, Albert; Ferrer Llop, Jose; Peña Carrera, Marta
    Linear algebra and its applications
    Date of publication: 2004-03
    Journal article

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  • Estructuras geométricas y deformaciones asociadas a sistemas lineales multivariables

     Puerta Sales, Fernando; Compta Creus, Albert; Magret Planas, Maria Dels Dolors; Clotet Juan, Jose; Puerta Coll, Francisco Javier; Peña Carrera, Marta; Ferrer Llop, Jose
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  • Stratification and bundle structure of the set of conditioned invariant subspaces in the general case

     Ferrer Llop, Jose; Puerta Sales, Fernando; Puerta Coll, Francisco Javier
    Systems & control letters
    Date of publication: 2003-01
    Journal article

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  • Miniversal deformations of marked matrices

     Compta Creus, Albert; Ferrer Llop, Jose; Puerta Sales, Fernando
    Linear algebra and its applications
    Date of publication: 2003-03
    Journal article

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  • Dimension of the Orbit of Marked Subspaces

     Compta Creus, Albert; Ferrer Llop, Jose; Peña Carrera, Marta
    Date: 2003-07
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  • Stratification and bundle structure of the set of conditioned invariant subspaces in the general case

     Ferrer Llop, Jose; Puerta Sales, Fernando; Puerta Coll, Francisco Javier
    Date: 2003-01
    Report

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  • Global reduction to the Kronecker canonical form of a Cr-family of time-invariant linear systems

     Puerta Coll, Francisco Javier; Puerta Sales, Fernando; Ferrer Llop, Jose
    Linear algebra and its applications
    Date of publication: 2002-02
    Journal article

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  • Matricial realizations of the solutions of the Carlson problem

     Compta Creus, Albert; Ferrer Llop, Jose
    Linear algebra and its applications
    Date of publication: 2002-09
    Journal article

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  • Global reduction to the Kronecker canonical form of a $C^r$-family of time-invariant linear systems

     Puerta Coll, Francisco Javier; Puerta Sales, Fernando; Ferrer Llop, Jose
    Date: 2002-04
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  • Miniversal Deformations of Marked Matrices

     Compta Creus, Albert; Ferrer Llop, Jose; Puerta Sales, Fernando
    Date: 2002-02
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    Versal deformations of invariant subspaces  Open access

     Ferrer Llop, Jose; Puerta Sales, Fernando
    Linear algebra and its applications
    Date of publication: 2001-08
    Journal article

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    We describe a miniversal deformation of invariant subspaces (with regard to a fixed endomorphism) by means of a technique which can be applied to obtain explicitly miniversal deformations in a general orbit space. In addition, we present an application to the problem of classifying invariant subspaces.

  • Contribució a l'estudi geomètric de subespais invariants respecte a transformacions i sistemes lineals  Open access

     Compta Creus, Albert
    Defense's date: 2001-10-19
    Department of Applied Mathematics I, Universitat Politècnica de Catalunya
    Theses

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    Mitjançant tècniques geomètriques, abordem les qüestions següents:(i) Estudi (caracterització, classificació, famílies diferenciables,...) d'una classe destacada de subespais invariants, els anomenats "marcats".(ii) Existència i construcció de solucions de l'anomenat problema de Carlson.(iii) Pertorbacions de matrius conservant un subespai invariant.I. Gohberg, P. Lancaster i L. Rodman defineixen una classe de subespais invariants, els marcats, com els que admeten una base de Jordan relativa a la restricció que sigui extensible a una base de Jordan de l'espai.J. Ferrer-F. Puerta-X. Puerta caracteritzen els subespais marcats en termes geomètrics i els classifiquen. Aquí, els caracteritzem de dues formes diferents: la primera utilitza la filtració doble de Jordan formada per les interseccions dels nuclis i les imatges de les potències de l'endomorfisme, i en particular retroba el resultat abans referit; la segona és en termes de la filtració triple, que resulta d'intersecar l'anterior amb les imatges de les potencies de la restricció, que permet generalitzar el teorema de classificació anterior.En relació amb la segona qüestió, recordem que el problema de Carlson consisteix en preguntar-se per l'existència d'una matriu amb una forma de Jordan determinada si són fixades les formes de Jordan d'un subespai invariant i del quocient. Mitjançant T. Klein es redueix el problema de Carlson a l'existència de les successions de Littlewood-Richardson. Recentment, com es pot veure en un article resum de W. Fulton, s'han trobat condicions a l'efecte. No obstant, no hi ha algorismes per construir solucions explícites. Aquí presentem una demostració geomètrica constructiva del resultat anterior que permet un algorisme per a l'obtenció de solucions.Com una aplicació important, obtenim que, fixades les característiques de Segre del subespai i del quocient, totes les característiques de Segre compatibles tenen alguna realització en qualsevol entorn de les que corresponen a un subespai marcat. Resulta, doncs, que totes les solucions al problema de Carlson apareixen pertorbant les solucions marcades elementals.Això motiva que en la tercera part d'aquest treball estudiem les deformacions d'una matriu que deixa invariant un subespai. Apliquem les tècniques usades per V.I. Arnold per a matrius quadrades per estudiar les matrius del mateix tipus que li són properes. N'obtenim l'expressió implícita d'una deformació miniversal i l'apliquem per obtenir explícitament una deformació miniversal d'una matriu marcada.Els dos primers problemes els tractem també per al cas de sistemes lineals, representats per parelles horitzontals de matrius (A,B). Per dualitat, és equivalent considerar parelles verticals, habitualment escrites (C,A), les quals es poden tractar com a aplicacions lineals definides en un subespai.I. Gohberg, P. Lancaster i L. Rodman estenen la definició de subespai invariant per una parella de matrius. Els subespais (C,A)-invariants també reben el nom de subespais invariants condicionats.Un subespai invariant condicionat es diu marcat si existeix una base de Brunovsky relativa a la restricció extensible a una base de Brunovsky del total. Obtenim una caracterització geomètrica dels subespais (C,A)-marcats, una família completa d'invariants que els classifiquen i condicions suficients per a la existència d'una base global de Brunovsky per a una família diferenciable de subespais (C,A)-marcats.El problema de Carlson també es generalitza de forma natural a parelles de matrius. Aquí, demostrem un teorema, anàleg al fet en el cas quadrat, quan la parella és observable i el quocient és un endomorfisme amb un sol valor propi. Aquest últim problema també ha estat resolt per I. Baragaña i I. Zaballa usant mètodes matricials. És remarcable que una relació directa entre les particions que caracteritzen els blocs de les matrius, que en el cas quadrat és solament necessària, és suficient per a garantir l'existència de solucions en aquest cas. Igualment generalitzem l'algorisme per a l'obtenció explícita de solucions.

    We study the next items geometrically:(iv) Characterization, classification, differentiable families,...of an important type of invariant subspaces called marked subspaces.(v) Existence and construction of Carlson problem solutions. (vi) Deformations of matrices that preserve a subspace.(vii) I. Gohberg, P. Lancaster and L. Rodman define the marked subspaces as the ones that have a Jordan basis of the restriction extensible to a Jordan basis of the whole space.J. Ferrer, F. Puerta, and X. Puerta characterize the marked subspaces geometrically and classify them. Here, we characterize them in two different forms: the first one uses the Jordan double filtration formed by kernels and images intersections of the powers of the endomorphism, and in particular find the above result again; the second one is in terms of the triple filtration formed by the intersection of the sets of the above filtration with the images of the powers of the restriction, that allows us to generalize the above classification theorem. In relation to the second item, we recall that the Carlson problem consists of asking by the existence of a matrix with a certain Jordan form if the Jordan form of an invariant subspace and his quotient are fixed. T. Klein reduces the Carlson problem to the existence of the Littlewood-Richardson sequences. Recently, as we can see in a W. Fulton summary paper, existence conditions for them are obtained. However there are not algorithms to construct explicit solutions. Here we present a geometrical proof of the above result that allows us an algorithm for that.As an important application, we obtain that, if the Segre characteristic of the subspace and the quotient are fixed, all the compatible Segre characteristics can be realized in a neighbourhood of some realization corresponding to a marked subspace. It follows that all the Carlson problem solutions appear perturbing the marked solutions.This fact causes that we study the deformations of matrices preserving a subspace in the third part of this paper. We apply the techniques used by V.I. Arnold in the study of deformations of square matrices to study the same kind of matrices that are near them. We obtain the explicit form of a miniversal deformation of a marked matrix.We also study the two first items in the linear system case, done by horizontal pairs of matrices (A,B). By duality, it is equivalent to considerate vertical pairs written habitually (C,A), that we can see as linear maps defined in a subspace.I. Gohberg, P. Lancaster and L. Rodman give the definition of an invariant subspace by a pair of matrices. The (C,A)-invariant subspaces are also known as conditioned invariant subspaces.We say that a conditioned invariant subspace is marked if there is a Brunovsky basis of the restriction extendible to a Brunovsky basis of the whole space. We obtain a geometrical characterization of (C,A)-marked subspaces, a complete family of invariants and sufficient conditions in order to guarantee the existence of a global Brunovsky basis of a differentiable family of (C,A)-marked subspaces.We can also generalize the Carlson problem for pairs of matrices in a natural way. Here, we prove a theorem, similar to the one for the square case, when the pair is observable and the quotient is an endomorphism with an only eigenvalue. I. Baragaña and I. Zaballa also solved this problem using matricial methods. We want to note that a direct relation between the partitions that characterize the blocks of the matrices is sufficient to guarantee the existence of solutions while it is only necessary in the square case. Also we generalize the algorithm to obtain explicit solutions.

  • Miniversal Deformations of Marked Matrices

     Ferrer Llop, Jose
    Tercer encuentro plenario: Estudio de la Estructura de Sistemas y Matrices
    Presentation's date: 2001-09-18
    Presentation of work at congresses

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  • Deformaciones De Subespacios Invariantes

     Ferrer Llop, Jose
    Tercer encuentro plenario: Estudio de la Estructura de Sistemas y Matrices
    Presentation's date: 2001-09-18
    Presentation of work at congresses

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