Perturbations preserving conditioned invariant subspaces

Compta, Albert; Ferrer, Josep; Peña, Marta

doi:10.1002/mma.1553

Cited by 1 publication

(5 citation statements)

References 17 publications

(41 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Thus, applying these techniques, one obtains the main result in , valid for a generic pair

scriptA MathClass-rel\in {scriptM}^{MathClass-bin*} MathClass-rel\subset scriptM

: Theorem Let

scriptA MathClass-rel= (C MathClass-punc, A) MathClass-rel\in M_{n MathClass-bin+ m MathClass-punc, n} (double-struckC)

be a pair of matrices and

S MathClass-rel\subset {double-struckC}^{n}

a d ‐dimensional ( C , A )‐invariant subspace:

scriptA MathClass-rel= ((), \begin{matrix} falsenonefalsearray \\ array A \\ array C \end{matrix}) MathClass-rel= ((), \begin{matrix} falsenonefalsearray \\ array A_{1} & array A_{3} \\ array 0 & array A_{2} \\ array C_{1} & array C_{2} \end{matrix})

Assume

scriptA MathClass-rel\in {scriptM}^{MathClass-bin*}

; that is to say, rank C 1 = r * = min( d , m ). Then, the following statements hold.…”

Section: Miniversal Deformation Preserving a (Ca)‐invariant Subspacementioning

confidence: 99%

“…In [13], one characterizes (Proposition 8 in [13]) the elements of the set M. Moreover, one shows (Theorem 10 in [13]) that these elements form a stratified manifold and that the stratum M is an open dense subset in M. Finally, one describes (Proposition 12 in [13]) a Lie group that acts on M in such a way that the orbits are just the equivalence classes in Definition 5.…”

Section: Definitionmentioning

confidence: 99%

“…Thus, applying these techniques, one obtains the main result in [13], valid for a generic pair A 2 M M:…”

Section: Theorem 11mentioning

confidence: 99%

“…The organization of this paper is as follows. In Sections 2 and 3, we recall some results obtained in regarding the differentiable structure of

scriptM

and the implicit form of a miniversal deformation of a pair

scriptA MathClass-rel\in {scriptM}^{MathClass-bin*}

, respectively. In Section 4, we solve the aforementioned equations when the pair is observable, and the subspace is marked (Theorem ), also obtaining a second miniversal deformation without repeated parameters (Theorem ).…”

Section: Introductionmentioning

confidence: 99%

“…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 . Some applications are derived: computation of the dimension of the orbits, characterization of structurally stable objects, study of bifurcations diagrams and others.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations