On (A, B)t-invariant subspaces having extendible Brunovsky bases

Compta, Albert; Ferrer, Josep

doi:10.1016/s0024-3795(96)00001-8

Cited by 13 publications

(12 citation statements)

References 5 publications

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“…We prove that this definition is equivalent to the one given in Definition .

\begin{matrix} aligned \\ right f MathClass-open(S MathClass-close) \cap Y & left = \{x \in Y : x = f MathClass-open(w MathClass-close), w \in S\} = \{(\begin{matrix} y \\ 0 \end{matrix}) : (\begin{matrix} y \\ 0 \end{matrix}) = (\begin{matrix} Aw \\ Cw \end{matrix}), w \in S\} = \{(\begin{matrix} Aw \\ 0 \end{matrix}) : Cw = 0, w \in S\} & right \\ right & left = \{Aw : w \in S \cap Ker C\} = A MathClass-open(S \cap Ker C MathClass-close) . \end{matrix}

Now, the characterization (b) is precisely the one obtained in [, Proposition 3.1] for the notion of f ‐invariant subspace defined above.

(b) 虠 (c)

The elements of

{double-struckC}^{d} MathClass-bin\cap Ker C

have the form

((), \begin{matrix} falsefalsearray \\ arraycenter x \\ arraycenter 0 \end{matrix})

with C 1 x = 0.…”

Section: Pairs Of Matrices Having a (Ca)‐invariant Fixed Subspacementioning

confidence: 80%

“…In any basis of X adapted to Y ⊂ X , Y is being identified with

((), \begin{matrix} falsefalsearray \\ arraycenter Y \\ arraycenter 0 \end{matrix})

, and the matrix of f is a pair

((), \begin{matrix} falsefalsearray \\ arraycenter A \\ arraycenter C \end{matrix})

where

A MathClass-rel\in M_{n} (double-struckC)

and

C MathClass-rel\in M_{m MathClass-punc, n} (double-struckC)

. In [, Definition 3.1], S is said to be f invariant if f ( S ) ∩ Y ⊂ S . We prove that this definition is equivalent to the one given in Definition .…”

Section: Pairs Of Matrices Having a (Ca)‐invariant Fixed Subspacementioning

confidence: 99%

See 1 more Smart Citation

Perturbations preserving conditioned invariant subspaces

Compta

Ferrer

Peña

2011

Math Methods in App Sciences

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“…We prove that this definition is equivalent to the one given in Definition .

\begin{matrix} aligned \\ right f MathClass-open(S MathClass-close) \cap Y & left = \{x \in Y : x = f MathClass-open(w MathClass-close), w \in S\} = \{(\begin{matrix} y \\ 0 \end{matrix}) : (\begin{matrix} y \\ 0 \end{matrix}) = (\begin{matrix} Aw \\ Cw \end{matrix}), w \in S\} = \{(\begin{matrix} Aw \\ 0 \end{matrix}) : Cw = 0, w \in S\} & right \\ right & left = \{Aw : w \in S \cap Ker C\} = A MathClass-open(S \cap Ker C MathClass-close) . \end{matrix}

Now, the characterization (b) is precisely the one obtained in [, Proposition 3.1] for the notion of f ‐invariant subspace defined above.

(b) 虠 (c)

The elements of

{double-struckC}^{d} MathClass-bin\cap Ker C

have the form

((), \begin{matrix} falsefalsearray \\ arraycenter x \\ arraycenter 0 \end{matrix})

with C 1 x = 0.…”

Section: Pairs Of Matrices Having a (Ca)‐invariant Fixed Subspacementioning

confidence: 80%

“…In any basis of X adapted to Y ⊂ X , Y is being identified with

((), \begin{matrix} falsefalsearray \\ arraycenter Y \\ arraycenter 0 \end{matrix})

, and the matrix of f is a pair

((), \begin{matrix} falsefalsearray \\ arraycenter A \\ arraycenter C \end{matrix})

where

A MathClass-rel\in M_{n} (double-struckC)

and

C MathClass-rel\in M_{m MathClass-punc, n} (double-struckC)

. In [, Definition 3.1], S is said to be f invariant if f ( S ) ∩ Y ⊂ S . We prove that this definition is equivalent to the one given in Definition .…”

Section: Pairs Of Matrices Having a (Ca)‐invariant Fixed Subspacementioning

confidence: 99%

Perturbations preserving conditioned invariant subspaces

Compta

Ferrer

Peña

2011

Math Methods in App Sciences

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“…In particular, P (or f ) is observable if and only if [4]). Let us see that the special form of P in section 3 appears in a natural way when invariant subspaces are considered.…”

Section: Proof Of the Equivalence (I) And (I ) And (Ii) And (Ii ) Tmentioning

confidence: 99%

A Geometric Approach to the Carlson Problem

Compta¹,

Ferrer²

2000

SIAM J. Matrix Anal. & Appl.

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“…With basis on the results in that section the two main goals of this paper are reached: the set of solutions of the generalized partial realization of a given nice sequence is provided with the desired stratified differentiable structure (Sect. 7), and the relationship between this structure and that of the cover problem obtained in [28] is clarified (Sect. 8).…”

Section: Introductionmentioning

confidence: 99%

“…h 4 F}, then = (3, 1, 0, 2, 0, 0), m = ((1, 2), (1, 4), (1, 4)), p = ((1,2), (1, 2, 4, 7),(1,2,4,7,9,12)),t = ((3),(2,3,5), (2, 3, 5, 6)), n = ((3),(2,5),(6),…”

mentioning

confidence: 99%

On the geometry of the generalized partial realization problem

Baragaña

Puerta²,

Puerta³

et al. 2010

Math. Control Signals Syst.

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The geometry of the set of generalized partial realizations of a finite nice sequence of matrices is studied. It is proved that this set is a stratified manifold, the dimension of their strata is computed and its connection with the geometry of the cover problem is clarified. The results can be applied, as a particular case, to the classical partial realization problem.Postprint (published version

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On (A, B)t-invariant subspaces having extendible Brunovsky bases

Cited by 13 publications

References 5 publications

Perturbations preserving conditioned invariant subspaces

Perturbations preserving conditioned invariant subspaces

A Geometric Approach to the Carlson Problem

On the geometry of the generalized partial realization problem

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