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 monic Hermitian determinantal representation as well as monic symmetric determinantal representation of size 2 for a given quadratic polynomial. Further we propose a method to construct such a monic determinantal representation (MDR) of size 2 if it exists. It is known that a quadratic polynomial f(x)=xTAx+bTx+1 has a symmetric MDR of size n+1 if A is 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. Finally, we characterize all quadratic polynomials that exhibit MDRs of any size.