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.
|Translated title of the contribution||On the intersection of null spaces for matrix substitutions in a non-commutative rational formal power series|
|Number of pages||6|
|Journal||Comptes Rendus Mathematique|
|State||Published - 15 Oct 2004|