A Complete Characterization of Quadratic Polynomials that are determinants of Linear Matrix Polynomials

Papri Dey, Harish K. Pillai

Published 2016-06-16Version 1

The problem of expressing a multivariate polynomial as the determinant of a monic (definite) symmetric (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 symmetric determinantal representation (MSDR) of size 2 for a given quadratic polynomial. Further if an MSDR exists we propose a method to construct such a $2 \times 2$ MSDR. It is known that a quadratic polynomial $f(x) = x^TAx + b^T x + 1$ has an MSDR 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 MSDR of size greater than 2, then it has an MSDR of size 2 too. We also characterize quadratic polynomials which exhibit diagonal MSDRs.