Permanents and Lorentzian time-semidefinite matrices Original Research Article

A hermitian matrix is said to be Lorentzian time-semidefinite if it has nonnegative diagonal elements and at most one positive eigenvalue. We show, using a generalization of a theorem of Grace due to Lars Hörmander that such matrices have nonnegative permanent. We establish a quantitative version of Hörmanderʹs result and apply that result to give an analogue of Hadamardʹs inequality for Lorentzian time-semidefinite matrices. We also consider some related generalized matrix functions inequalities.