2020/10/15 by Anthony Stefan, Stefan, Anthony, Aaron Welters +1
Computer Science · #15A22 #32A08 #47A13 #47A56 #93B25 #93B28 #Complex Variables (math.CV) #FOS: Mathematics #Matrix Theory and Algorithms
paper · pdf · doi:10.48550/arxiv.2010.08088
openalex publication_date 2020/10/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove a realization theorem for rational functions of several complex\nvariables which extends the main theorem of M. Bessmertnyi, "On realizations of\nrational matrix functions of several complex variables," in Vol. 134 of Oper.\nTheory Adv. Appl., pp. 157-185, Birkh "auser Verlag, Basel, 2002. In contrast\nto Bessmertnyi's approach of solving large systems of linear equations, we use\nan operator theoretical approach based on the theory of Schur complements. This\nleads to a simpler and more "natural" construction to solving the realization\nproblem as we need only apply elementary algebraic operations to Schur\ncomplements such as sums, products, inverses, and compositions. A novelty of\nour approach is the use of Kronecker product as opposed to the matrix product\nin the realization problem. As such our synthetic approach leads to a solution\nof the realization problem that has potential for further extensions and\napplications within multidimensional systems theory especially for those linear\nmodels associated with electric circuits, networks, and composites.\n