Order intervals of matrices and linear transformations

Cones and partial orders have been topics of independent study. We explore in detail the relationship between cones and partial orders. The idea of order intervals and the gauge function are discussed. When used together these allow us to generate norms on the underlying vector spaces which are in some sense "consistent" with the partial order.; Next, the theory of order intervals is applied to matrix spaces. In particular, characterizations of some order intervals and the resulting norms are given for the cone of nonnegative matrices in the real vector space of real matrices and the cone of positive semi-definite matrices in the real vector space of hermitian matrices. Finally, order intervals on the cone of positive semi-definite preserving maps in the real space of hermitian preserving linear transformations are considered as well as order intervals on subcones of this cone.