So assume that D is the zero matrix. In a next step only the matrices remain to be considered. Thus the proof is completed. After investigating the set we are ready to look at the linear subspace L of the cone of Kalmanson matrices. Note that .
L is necessary.
Moreover, it is easy to verify that all matrices of are lineraly independent in and that . Due to corollary 2.4 the set constitutes a basis of the linear subspace L . The next theorem gives the additive characterization of Kalmanson matrices. The proof is done by applying Theorem 2.1. First of all, due to Lemma 2.5 the set constitutes a basis of the linear subspace L .