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In this paper we explore determinantal representations of multiaffinepolynomials and consequences for the image of various spaces of matrices underthe principal minor map. We show that a real multiaffine polynomial has adefinite Hermitian determinantal representation if and only if all of itsso-called Rayleigh differences factor as Hermitian squares and use thischaracterization to conclude that the image of the space of Hermitian matricesunder the principal minor map is cut out by the orbit of finitely manyequations and inequalities under the action of $({\rm SL}_2(\mathbb{R}))^{n}\rtimes S_{n}$. We also study such representations over more general fieldswith quadratic extensions. Factorizations of Rayleigh differences prove aneffective tool for capturing subtle behavior of the principal minor map. Incontrast to the Hermitian case, we give examples to show for any field$\mathbb{F}$, there is no finite set of equations whose orbit under $({\rmSL}_2(\mathbb{F}))^{n} \rtimes S_{n}$ cuts out the image of $n\times n$matrices over $\mathbb{F}$ under the principal minor map for every $n$.

Determinantal representations and the image of the principal minor map