Schur Complement Algebra and Operations with Applications in Multivariate Functions, Realizations, and Representations
Stefan, Anthony Dean
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We provide a new approach to the following multidimensional realizability problem: Can an arbitrary square matrix, whose entries are from the field of multivariate rational functions over the complex numbers, be realized as a Schur complement of a linear matrix pencil with symmetries? To answer this problem, we prove the main theorem of M. Bessmertnyı̆,“On realizations of rational matrix functions of several complex variables,” in Vol. 134 of Oper. Theory Adv. Appl., pp. 157-185, Birkhäuser Verlag, Basel, 2002 and have included additional symmetries as an extension to his results. Furthermore, we were so thorough in our constructive approach that we also prove every real multivariate polynomial has a symmetric determinantal representation, which was first proved in J. W. Helton, S. A. McCullough, and V. Vinnikov, Noncommutative convexity arises from linear matrix inequalities, J. Funct. Anal. 240, 105-191 2006, as a direct application of our techniques. Our perspective is from a more “natural” and algorithmic approach using Schur complement algebra and operations with algebraic-functional symmetries. To further motivate the use of the Schur complement in realizability theory and its applications to synthesis and inverse problems, we give three quintessential examples of the Schur complement with symmetries in multivariate applied models: impedance matrices, Dirichlet-to-Neumann map, and effective conductivity tensor in the theory of composites. We then conclude with a discussion on the open problems related to and future directions of our study.