A Square Departure From Symmetry in Matrix Cones

Conic optimization problems are usually understood to be problems over some cone of symmetric matrices like the semidefinite or the copositive matrix cone. In this note, we investigate the changes that have to be made when moving from symmetric to nonsymmetric matrices. We introduce the proper definitions and study the dual of a cone of nonsymmetric matrices. Next, we attempt to generalize the well known concept of cp-rank to nonsymmetric matrices. Finally, we derive some new results on symmetric and nonsymmetric copositive-plus matrices.