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 representtaion (MDR) of size 2 if it exists. It is known that a quadratic polynomial f (x) = x T Ax + b T x + 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. We also characterize quadratic polynomials which exhibit diagonal MDRs.

In this paper, we consider the problem of representing a multivariate polynomial as the determinant of a definite (monic) symmetric/Hermitian linear matrix polynomial (LMP). Such a polynomial is known as determinantal polynomial. Determinantal polynomials can characterize the feasible sets of semidefinite programming problems that motivates us to deal with this problem. We introduce the notion of generalized mixed discriminant of matrices which translates the determinantal representation problem into computing a point of a real variety of a specified ideal. We develop an algorithm to determine such a determinantal representation of a bivariate polynomial of degree d. Then we propose a heuristic method to obtain a monic symmetric determinantal representation of a multivariate polynomial of degree d.

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Determinantal polynomials play a crucial role in semidefinite programming problems. Helton-Vinnikov proved that real zero (RZ) bivariate polynomials are determinantal. However, it leads to a challenging problem to compute such a determinantal representation. We provide a necessary and sufficient condition for the existence of definite determinantal representation of a bivariate polynomial by identifying its coefficients as scalar products of two vectors where the scalar products are defined by orthostochastic matrices. This alternative condition enables us to develop a method to compute a monic symmetric/Hermitian determinantal representations for a bivariate polynomial of degree d. In addition, we propose a computational relaxation to the determinantal problem which turns into a problem of expressing the vector of coefficients of the given polynomial as convex combinations of some specified points. We also characterize the range set of vector coefficients of a certain type of determinantal bivariate polynomials.AMS Classification (2010). 15A75, 15B10, 15B51, 90C22.

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