A Complete Characterization of Determinantal Quadratic Polynomials

The problem of expressing a multivariate polynomial as the determinant of a monic (definite) symmetric or Hermitian linear matrix polynomial (LMP) has drawn a huge amount of attention due to its connection with optimization problems. In this paper we provide a necessary and sufficient condition for the existence of \textit{monic Hermitian determinantal representation} as well as \textit{monic symmetric determinantal representation} of size $2$ for a given quadratic polynomial. Further we propose a method to construct such a monic determinantal representtaion (MDR) of size $2$ if it exists. It is known that a quadratic polynomial $f(\x)=\x^{T}A\x+b^{T}\x+1$ has a symmetric MDR of size $n+1$ if $A$ is \textit{negative semidefinite}. We prove that if a quadratic polynomial $f(\x)$ with $A$ which is not negative semidefinite has an MDR of size greater than $2$, then it has an MDR of size $2$ too. We also characterize quadratic polynomials which exhibit diagonal MDRs.