Determinantal Representations and the Image of the Principal Minor Map

Algebra Student Seminar
Friday, December 9, 2022 - 10:00am for 1 hour (actually 50 minutes)
Skiles 006
Abeer Al Ahmadieh – Georgia Institute of Technology –
Kevin Shu

 The principal minor map takes an n  by n square matrix to the length 2^n-vector of its principal minors. A basic question is to give necessary and sufficient conditions that characterize the image of various spaces of matrices under this map. In this talk, I will describe the image of the space of complex matrices using a characterization of determinantal representations of multiaffine polynomials, based on the factorization of their Rayleigh differences. Using these techniques I will give equations and inequalities characterizing the images of the spaces of real and complex symmetric and Hermitian matrices. For complex symmetric matrices, this recovers a result of Oeding from 2011. If time permits, I will also give examples to prove that for general matrices no such finite characterization is possible. This is based on joint work with Cynthia Vinzant.