[Sur l'intersection des noyaux des opérateurs de substitution de matrices pour les séries formelles non commutatives.]
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 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.
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@article{CRMATH_2004__339_8_533_0,
author = {Alpay, Daniel and Kalyuzhny{\i}̆-Verbovetzki{\i}̆, Dmitry S.},
title = {On the intersection of null spaces for matrix substitutions in a non-commutative rational formal power series},
journal = {Comptes Rendus. Math\'ematique},
pages = {533--538},
publisher = {Elsevier},
volume = {339},
number = {8},
year = {2004},
doi = {10.1016/j.crma.2004.08.005},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2004.08.005/}
}
TY - JOUR AU - Alpay, Daniel AU - Kalyuzhnyı̆-Verbovetzkiı̆, Dmitry S. TI - On the intersection of null spaces for matrix substitutions in a non-commutative rational formal power series JO - Comptes Rendus. Mathématique PY - 2004 SP - 533 EP - 538 VL - 339 IS - 8 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2004.08.005/ DO - 10.1016/j.crma.2004.08.005 LA - en ID - CRMATH_2004__339_8_533_0 ER -
%0 Journal Article %A Alpay, Daniel %A Kalyuzhnyı̆-Verbovetzkiı̆, Dmitry S. %T On the intersection of null spaces for matrix substitutions in a non-commutative rational formal power series %J Comptes Rendus. Mathématique %D 2004 %P 533-538 %V 339 %N 8 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2004.08.005/ %R 10.1016/j.crma.2004.08.005 %G en %F CRMATH_2004__339_8_533_0
Alpay, Daniel; Kalyuzhnyı̆-Verbovetzkiı̆, Dmitry S. On the intersection of null spaces for matrix substitutions in a non-commutative rational formal power series. Comptes Rendus. Mathématique, Tome 339 (2004) no. 8, pp. 533-538. doi : 10.1016/j.crma.2004.08.005. http://www.numdam.org/articles/10.1016/j.crma.2004.08.005/
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