Evaluating approximations of the semidefinite cone with trace normalized distance

We evaluate the dual cone of set diagonally dominant matrices (resp., scaled matrices), namely $$\mathcal{DD}_n^*$$ $$\mathcal{SDD}_n^*$$ ), as an approximation semidefinite cone. prove that norm normalized distance, proposed by Blekherman et al. [5], between a $$\mathcal{S}$$ and has same value whenever $$\mathcal{SDD}_n^* \subseteq \mathcal{S} \mathcal{DD}_n^*$$ . This implies distance is not sufficient measure to these approximations. As new compensate for weakness we propose called trace distance. $$\mathcal{S}^n_+$$ different from one give exact values distances.