The Lyapunov order for real matrices

The real Lyapunov order in the set of real n×n matrices is a relation defined as follows: A⩽B if, for every real symmetric matrix S, SB+BtS is positive semidefinite whenever SA+AtS is positive semidefinite. We describe the main properties of the Lyapunov order in terms of linear systems theory, Nevenlinna–Pick interpolation and convexity.