Directed Partial Orders of Matrices
by
Piotr J. Wojciechowski
The University of Texas at El Paso

Certain directed partial orders of matrices appear naturally in linear algebra and its applications. Besides the lattice orders, the best studied ones are the orders whose positive cones are the sets \Pi(O), of all matrices preserving a regular cone O in an n-dimensional Euclidean space. Vast literature on this subject includes works by H. Schneider, G. P. Barker and R. Loewy.

It will be shown that O is essentially the only \Pi(O)-invariant cone. Consequently, we obtain a characterization of all maximal directed partial orders on the n×n matrix algebra: P is maximal if and only if P=\Pi(O) for some regular cone O. The method used in the proof involves a concept of simplicial separation, allowing a regular cone to be separated from an outside point by means of a simplicial cone.

Date received: January 28, 2002

Copyright © 2002 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cain-05.