Cones of diagonally dominant matrices

George Phillip Barker and David Hilding Carlson

Vol. 57 (1975), No. 1, 15–32

DOI: 10.2140/pjm.1975.57.15

Abstract

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 H↦AH + HA^∗ determined by a positive stable matrix A.

Mathematical Subject Classification 2000

Primary: 15A60

Milestones

Received: 27 February 1974

Revised: 25 October 1974

Published: 1 March 1975