The left invariant metric in the general linear group

TL;DR

This paper investigates left invariant p-norm induced metrics on the general linear group, establishing existence, uniqueness, and explicit forms of geodesics, with extensions to compact operators and spectral analysis.

Contribution

It provides a comprehensive analysis of extremal paths and geodesics under p-norm metrics on the general linear group, including explicit solutions and spectral properties.

Findings

Geodesics have constant singular values and multiplicity.

Riemannian geodesics are products of one-parameter groups.

Geodesics are one-parameter groups iff initial velocity is normal.

Abstract

Left invariant metrics induced by the p-norms of the trace in the matrix algebra are studied on the general lineal group. By means of the Euler-Lagrange equations, existence and uniqueness of extremal paths for the length functional are established, and regularity properties of these extremal paths are obtained. Minimizing paths in the group are shown to have a velocity with constant singular values and multiplicity. In several special cases, these geodesic paths are computed explicitly. In particular the Riemannian geodesics, corresponding to the case p=2, are characterized as the product of two one-parameter groups. It is also shown that geodesics are one-parameter groups if and only if the initial velocity is a normal matrix. These results are further extended to the context of compact operators with p-summable spectrum, where a differential equation for the spectral projections of the velocity vector of an extremal path is obtained.