A Short Proof of the Symmetric Determinantal Representation of Polynomials

We provide a short proof of the theorem that 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 (2006), 105-191. We then provide an example using our approach and extend our results from the real field $\mathbb{R}$ to an arbitrary field $\mathbb{F}$ different from characteristic $2$. The new approach we take is only based on elementary results from the theory of determinants, the theory of Schur complements, and basic properties of polynomials.