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

Abstract

Abstract Consider the Lie group of n × n complex unitary matrices U ( n ) endowed with the bi-invariant Finsler metric given by the spectral norm, ‖ X ‖ U = ‖ U ⁎ X ‖ ∞ = ‖ X ‖ ∞ for any X tangent to a unitary operator U. Given two points in U ( n ) , in general there exist infinitely many curves of minimal length. In this paper we provide a complete description of such curves and as a consequence we give an equivalent condition for uniqueness. Similar studies are done for the Grassmann manifolds. On the other hand, consider the cone of n × n positive invertible matrices G l ( n ) + endowed with the bi-invariant Finsler metric given by the trace norm, ‖ X ‖ 1 , A = ‖ A − 1 / 2 X A − 1 / 2 ‖ 1 for any X tangent to A ∈ G l ( n ) + . In this context, also exist infinitely many curves of minimal length. In this paper we provide a complete description of such curves proving first a characterization of the minimal curves joining two Hermitian matrices X , Y ∈ H ( n ) . The last description is also used 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.