F. Puerta

Differentiable families of subspaces

Abstract We show how the differentiable families of subspaces can be studied, from a geometrical point of view, by means of their representation as a differentiable mapping from the manifold of parameters M into the Grassmann manifold Gr k,n of all k -dimensional subspaces of K n , and the consideration of principal bundles over Gr k,n and M . In order to achieve this, first we present a survey of the topology, the differentiable structure and the bundle structure of Gr k,n . In particular, we show that the topology induced by the gap metric is equivalent to the usual quotient topology. Second, we use these structures to prove two local characterizations of the differentiability of a family of subspaces, and we state that these characterizations can be globalized if M is contractible. Finally, we apply these properties to study the existence of global Jordan basis of class C r .

Brunowsky local form of a holomorphic family of pairs of matrices

Following Arnold’s techniques, we obtain a local canonical form of a holomorphic family of pairs of matrices ( A( A), B(h)) ac t e d on by the state feedback group. We obtain an explicit formula to compute the dimension of the base space of any miniversal deformation of ( A(O), B(O)). W e m ak e some applications to local perturbations of a pair of matrices.

Differentiable structure of the set of controllable (A,B)t-invariant subspaces☆

Abstract Given (A,B) t ∈ Hom( C n , C n+m ) observable, we prove that the set of ( A,B ) t -invariant subspaces having a fixed Brunovsky-Kronecker structure is a connected manifold, and we compute its dimension. Also, we include some applications of these results.

Similarity of non-everywhere defined linear maps☆

Abstract The equivalence between linear maps defined only on a subspace (or, by duality, defined modulo a subspace) is studied, and applied to block similarity of rectangular matrices. In that way, we find the complete families of invariants obtained by Brunovsky and Zaballa, and the results of Gohberg and others for this particular case. Also, we describe explicit methods for obtaining the reduced matrix and the corresponding bases.

GEOMETRIC CHARACTERIZATION AND CLASSIFICATION OF MARKED SUBSPACES

Abstract Given an endomorphism of a finite dimensional C -vector space, we obtain geometric conditions in order to characterize and to classify the marked invariant subspaces W ⊂ E (that is to say, the invariant subspaces having some Jordan basis which can be extended to a Jordan basis of E ). Both conditions are expressed in terms of the double family of subspaces (Ker f h ∩ Im f d ). The starting point is a new criterion for extensibility of Jordan bases.

Versal deformations of invariant subspaces

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.

Regularity of the Brunovsky--Kronecker Stratification

We study the partition of the set of pairs of matrices according to the Brunovsky--Kronecker type. We show that it is a constructible stratification, and that it is Whitney regular when the second matrix is a column matrix. We give an application to the obtainment of bifurcation diagrams for few-parameter generic families of linear systems.

Miniversal deformations of marked matrices

Abstract Given the set of square matrices M ⊂M n+m (C) that keep the subspace W = C n x {0}⊂ C n + m invariant, we obtain the implicit form of a miniversal deformation of a matrix a∈ M , and we compute it explicitely when this matrix is marked (this is, if there is a permutation matrix p ∈ M n + m ( C ) such that p −1 ap is a Jordan matrix). We derive some applications to tackle the classical Carlson problem.

On the geometry of the set of controllability subspaces of a pair (A,B)

Abstract Given a controllable system defined by a pair of matrices (A,B), we investigate the geometry of the set of controllability subspaces. This set is a subset of the set of (A,B)-invariant subspaces. We prove that, in fact, it is a stratified submanifold and we compute its dimension.

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

Abstract We present an approach to the Kronecker equivalence of quadruples of matrices (A, B,C, D) based on a natural equivalence relation between pairs of linear mappings. We apply this approach to smooth families of quadruples of constant Kronecker type, obtaining smooth families of transformation matrices that pointwise reduce each quadruple to its Kronecker canonical form.