Characterizing principal minors of symmetric matrices via determinantal multiaffine polynomials

2021/05/27 by Abeer Al Ahmadieh, Cynthia Vinzant, Ahmadieh, Abeer Al +1
Mathematics · #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.2105.13444

Abstract

Here we consider the image of the principal minor map of symmetric matrices over an arbitrary unique factorization domain R. By exploiting a connection with symmetric determinantal representations, we characterize the image of the principal minor map through the condition that certain polynomials coming from so-called Rayleigh differences are squares in the polynomial ring over R. In almost all cases, one can characterize the image of the principal minor map using the orbit of Cayley's hyperdeterminant under the action of (SL₂(R))ⁿ \rtimes S_n. Over the complex numbers, this recovers a characterization of Oeding from 2011, and over the reals, the orbit of a single additional quadratic inequality suffices to cut out the image. Applications to other symmetric determinantal representations are also discussed.

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