Singularities of rational functions and minimal factorizations: The noncommutative and the commutative setting |
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Authors: | Dmitry S. Kaliuzhnyi-Verbovetskyi |
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Affiliation: | a Department of Mathematics, Drexel University, 3141, Chestnut Str., Philadelphia, PA 19104, United States b Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel |
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Abstract: | 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. |
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Keywords: | Primary: 16S38 Secondary: 12E15 15A54 93B20 93B55 |
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