A Complete Characterization of Determinantal Quadratic Polynomials

2016/06/16 by Dey, Papri, Pillai, Harish K. · 1 citation
#FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.1606.05184

Abstract

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^TA\x+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.

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