Comptes Rendus
no. 8
Algebra
On the intersection of null spaces for matrix substitutions in a non-commutative rational formal power series
[Sur l'intersection des noyaux des opérateurs de substitution de matrices pour les séries formelles non commutatives.]

Présenté par : Jean-Pierre Kahane

Daniel Alpay1 ; Dmitry S. Kalyuzhnyı̆-Verbovetzkiı̆1
1 Department of Mathematics, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel
Comptes Rendus. Mathématique, Volume 339 (2004) no. 8, pp. 533-538.

Nous démontrons une formule qui relie l'intersection des noyaux des coefficients d'une série formelle de N variables non commutatives à l'intersection des noyaux des valeurs de cette série pour des N-uplets de matrices d'ordre n pour n assez grand. Comme application, nous donnons des critères d'observabilité, de commandabilité et de minimalité en termes de représentations pour les séries formelles reconnaissables et en utilisant des substitutions matricielles.

For a rational formal power series in N non-commuting indeterminates, with matrix coefficients, we establish the formula which relates the intersection of the null spaces of coefficients to the intersection of the null spaces of values of this series at N-tuples of n×n matrices, for n large enough. As an application, we formulate the criteria of observability, controllability, and minimality for a recognizable formal power series representation in terms of matrix substitutions.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2004.08.005
Daniel Alpay 1 ; Dmitry S. Kalyuzhnyı̆-Verbovetzkiı̆ 1

1 Department of Mathematics, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel 
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Daniel Alpay; Dmitry S. Kalyuzhnyı̆-Verbovetzkiı̆. On the intersection of null spaces for matrix substitutions in a non-commutative rational formal power series. Comptes Rendus. Mathématique, Volume 339 (2004) no. 8, pp. 533-538. doi : 10.1016/j.crma.2004.08.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.08.005/

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