  We study the set of generalized principal $\mathfrak{g}$-logarithms of any
matrix belonging to a connected SVD-closed subgroup $G$ of $U_n$, with Lie
algebra $\mathfrak{g}$. This set is a non-empty disjoint union of a finite
number of subsets diffeomorphic to homogeneous spaces, and it is related to a
suitable set of minimizing geodesics. Many particular cases for the group $G$
are explicitly analysed.
