A short proof of the symmetric determinantal representation of polynomials

We provide a short proof of the theorem that every real multivariate polynomial has symmetric determinantal representation, which was first proved in Helton et al. (2006) [4] . then an example using our approach and extend results from field R to arbitrary F different characteristic 2. The new we take is only based on elementary theory determinants, Schur complements, basic properties polynomials.