Cones of diagonally dominant matrices

The extreme rays of several cones of complex and real diagonally dominant matrices, and their duals, are identified. Several results on lattices of faces of cones are given. It is then shown that the dual (in the real space of hermitian matrices) of the cone of hermitian diagonally dominant matrices cannot be the image of the cone of positive semidefinite matrices under any nonsingular linear transformation; in particular, it cannot be the image of the cone of positive semidefinite matrices under the Ljapunov transformation ΈH AH+ HA* determined by a positive stable matrix A.