Minimal curves in U(n) and Gl(n)+ with respect to the spectral and the trace norms

TL;DR

This paper characterizes minimal curves in the Lie groups U(n) and Gl(n)+ under spectral and trace norms, providing explicit descriptions, conditions for uniqueness, and convexity properties of intermediate points.

Contribution

It offers a complete description of minimal length curves in U(n) and Gl(n)+ with respect to spectral and trace norms, including uniqueness conditions and convexity of intermediate points.

Findings

Unique minimal curves exist when the spectrum of U*V is contained in a specific set.

The set of intermediate points is geodesically convex under certain conditions.

Explicit constructions of minimal paths in U(n) and Gl(n)+ are provided.

Abstract

Consider the Lie group of n x n complex unitary matrices U(n) endowed with the bi-invariant Finsler metric given by the spectral norm, ||X||_U = ||U*X||_{sp} = ||X||_{sp} for any X tangent to a unitary operator U. Given two points in U(n), in general there exists infinitely many curves of minimal length. The aim of this paper is to provide a complete description of such curves. As a consequence of this description, we conclude that there is a unique curve of minimal length between U and V if and only if the spectrum of U*V is contained in a set of the form \{e^{i \theta}, e^{-i \theta}\} for some \theta \in [0, \infty). Similar studies are done for the Grassmann manifolds. Now consider the cone of n x n positive invertible matrices Gl(n)+ endowed with the bi-invariant Finsler metric given by the trace norm, ||X||_{1, A} = ||A^{-1/2}XA^{-1/2}||_1 for any X tangent to A \in Gl(n)+. In this context, given two points A,B \in Gl(n)+ there exists infinitely many curves of minimal length. In order to provide a complete description of such curves, we provide a characterization of the minimal curves joining two Hermitian matrices X, Y \in H(n). As a consequence of the last description, we provide a way to construct minimal paths in the group of unitary matrices U(n) endowed with the bi-invariant Finsler metric ||X||_{1, U} = ||U*X||_{1} = ||X||_{1} for any X tangent to U \in U(n). We also study the set of intermediate points in all the previous contexts. Between two given unitary matrices U and V we prove that this set is geodesically convex provided ||U - V||_{sp} < 1. In Gl(n)+ this set is geodesically convex for every unitarily invariant norm.