  We evaluate the dual cone of the set of diagonally dominant matrices (resp.,
scaled diagonally dominant matrices), namely ${\cal DD}_n^*$ (resp., ${\cal
SDD}_n^*$), as an approximation of the semidefinite cone. We prove that the
norm normalized distance, proposed by Blekherman et al. (2022), between a set
${\cal S}$ and the semidefinite cone has the same value whenever ${\cal
SDD}_n^* \subseteq {\cal S} \subseteq {\cal DD}_n^*$. This implies that the
norm normalized distance is not a sufficient measure to evaluate these
approximations. As a new measure to compensate for the weakness of that
distance, we propose a new distance, called the trace normalized distance. We
prove that the trace normalized distance between ${\cal DD}_n^*$ and ${\cal
S}^n_+$ has a different value from the one between ${\cal SDD}_n^*$ and ${\cal
S}^n_+$ and give the exact values of these distances.
