SVD-closed subgroups of the unitary group: generalized principal logarithms and minimizing geodesics

TL;DR

This paper explores the structure of generalized principal logarithms and geodesics in certain subgroups of the unitary group, providing explicit analyses for specific cases and revealing their geometric properties.

Contribution

It characterizes the set of generalized principal logarithms for SVD-closed subgroups of the unitary group and relates them to minimizing geodesics, with explicit case analyses.

Findings

The set of generalized principal logarithms forms a union of homogeneous spaces.

These logarithms are connected to minimizing geodesics in the subgroup.

Explicit descriptions are provided for particular subgroup cases.

Abstract

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.