Abstract:
In the relatively new subject of semidefinite programming, positive definiteness plays the role of scalar positivity and inequalities in the PSD partial order replace standard inequalities, relative to linear programming. The subject has had great success in solving or approximately solving hard problems, and it has also presented interesting new mathematical problems about positive definite matrices. One of these deals with maintenance of positive semidefiniteness under 0-ing out some off-diagonal entries ("gangster operators"). We discuss some recent results in this area.