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

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 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, 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.