Miniversal deformations of marked matrices

Miniversal deformations of observable marked matrices

“…In the context of versal deformations, it seems natural to consider the situation when both a matrix and an invariant subspace are involved and both or one of the elements of this couple is perturbed. So, in , the perturbation of a square matrix preserving an invariant subspace is studied. In particular, this perturbation gives all the solutions of the Carlson problem, and hence, explicit realizations can be obtained (see ).…”

“…Here, we complete the generalization of to linear systems started in . That is to say, we consider the perturbation of a pair of matrices preserving a given conditioned invariant subspace: We obtain the explicit form of the miniversal deformation of a given pair when it is observable, and the subspace is marked (that is to say, when there is a Brunovsky–Kronecker basis (BK‐basis) of the subspace extendible to a BK‐basis of the whole space), by solving the equations obtained in .…”

Perturbations preserving conditioned invariant subspaces

“…Notice that for the particular case of square matrices in , one obtains simply . Hence, the generalization to pairs of matrices is non‐trivial.…”

“…In the context of versal deformations, it seems natural to consider the situation when both a matrix and an invariant subspace are involved and both or one of the elements of this couple is perturbed. So, in , the perturbation of a square matrix preserving an invariant subspace is studied. In particular, this perturbation gives all the solutions of the Carlson problem, and hence explicit realizations can be obtained .…”

“…Here, we generalize to linear systems. That is to say, we consider the perturbation of a pair of matrices preserving a given conditioned invariant subspace: we obtain the equations of a miniversal deformation (i.e., minimal dimensional versal deformation).…”