A Complete Characterization of Determinantal Quadratic Polynomials
Papri Dey, Harish K. Pillai

TL;DR
This paper characterizes when quadratic polynomials can be represented as determinants of small-sized monic Hermitian or symmetric linear matrix polynomials, providing conditions, construction methods, and characterizations of special cases.
Contribution
It offers a complete characterization and construction method for monic Hermitian and symmetric determinantal representations of quadratic polynomials of size 2.
Findings
Necessary and sufficient conditions for size 2 monic Hermitian and symmetric representations.
A method to construct such representations if they exist.
Quadratic polynomials with non-negative semidefinite matrices have size n+1 symmetric MDRs.
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 \textit{monic Hermitian determinantal representation} as well as \textit{monic symmetric determinantal representation} of size for a given quadratic polynomial. Further we propose a method to construct such a monic determinantal representtaion (MDR) of size if it exists. It is known that a quadratic polynomial has a symmetric MDR of size if is \textit{negative semidefinite}. We prove that if a quadratic polynomial with which is not negative semidefinite has an MDR of size greater than , then it has an MDR of size…
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