Minimal curves in U(n) and Gl(n)+ with respect to the spectral and the trace norms
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 \ei θ, e-i θ\ for some θ ∈ [0, ∞). 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-1/2XA-1/2||1 for any X tangent to A ∈ Gl(n)+. In this context, given two points A,B ∈ 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 ∈ 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 ∈ 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.