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

ANTEZANA, JORGE; GHIGLIONI, EDUARDO; STOJANOFF, DEMETRIO

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

ACADEMIC PRESS INC ELSEVIER SCIENCE

Año: 2020 vol. 483

0022-247X

Consider the Lie group of nxn complex unitary matrices U(n) endowed with the bi-invariant Finsler metric given by the spectral norm,|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 nxn positive invertible matrices P(n)endowed with the bi-invariant Finsler metric given by the trace norm,|X|_{1,A}=|A^{-1/2}XA^{-1/2}|_1for any X tangent to A in P(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 and Y. 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|_1for any X tangent to U in U(n). We also study the set of intermediate points in all the previous contexts.