Minimal curves in u(n) and gl(n)⁺ with respect to the spectral and the trace norms

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, parallel to X parallel to(U) =parallel to U*X parallel to(infinity) = parallel to X parallel to(infinity) 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 x n positive invertible matrices gl(n)(+) endowed with the bi-invariant Finsler metric given by the trace norm, parallel to X parallel to(1,A) =parallel to A(-1/2)XA(-1/2)parallel to(1) for any X tangent to A is an element of gl(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 is an element of 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 parallel to X parallel to(1,U )=parallel to U*X parallel to(1) =parallel to X parallel to(1) FOR any X tangent to U is an element of u(n). We also study the set of intermediate points in all the previous contexts. (C) 2019 Elsevier Inc. All rights reserved.