  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.
