Algebra
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.

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.

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.

Accepted:
Published online:
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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