Extrémales du cône des matrices de type non négatif, à coefficients positifs ou nuls

Abstract We are here concerned with the following problem: Describe the extremals of the convex cone of n × n matrices, of nonnegative type, with nonnegative coefficients. We first extract from a simple general geometric result about faces of convex sets, a necessary condition of extremality which solves the problem for n ⩽ 4. Then, using again the same geometrical result, we establish an algebraic criterion of extremality. We apply this criterion to matrices of order n and rank n − 2; this solves the problem for n = 5. We end with an explicit description of extremals for n = 6.