Nonnegative definite hermitian matrices with increasing principal minors

A nonzero nonnegative definite hermitian m by m matrix A has increasing principal minors if the value of each principle minor of A is not less than the value each of its subminors. For $m>1$ we show $A$ has increasing principal minors if and only if $A^{-1}$ exists and its diagonal entries are less or equal to 1.Comment: 3 page

Polynomials with Lorentzian Signature, and Computing Permanents via Hyperbolic Programming

We study the class of polynomials whose Hessians evaluated at any point of a closed convex cone have Lorentzian signature. This class is a generalization to the remarkable class of Lorentzian polynomials. We prove that hyperbolic polynomials and conic stable polynomials belong to this class, and the set of polynomials with Lorentzian signature is closed. Finally, we develop a method for computing permanents of nonsingular matrices which belong to a class that includes nonsingular $k$-locally singular matrices via hyperbolic programming