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Singularities of rational functions and minimal factorizations: the noncommutative and the commutative setting. (English) Zbl 1217.47032

Summary: We show that the singularities of a matrix-valued noncommutative rational function which is regular at zero coincide with the singularities of the resolvent in its minimal state space realization. The proof uses a new notion of noncommutative backward shifts. As an application, we establish the commutative counterpart of the singularities theorem: the singularities of a matrix-valued commutative rational function which is regular at zero coincide with the singularities of the resolvent in any of its Fornasini-Marchesini realizations with the minimal possible state space dimension. The singularities results imply the absence of zero-pole cancellations in a minimal factorization, both in the noncommutative and in the commutative setting.

MSC:

47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
47N70 Applications of operator theory in systems, signals, circuits, and control theory
93B20 Minimal systems representations
93B55 Pole and zero placement problems

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NCAlgebra
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