IAM   02674
INSTITUTO ARGENTINO DE MATEMATICA ALBERTO CALDERON
Unidad Ejecutora - UE
artículos
Título:
The left invariant metric in the general linear group
Autor/es:
ANDRUCHOW, E.; LAROTONDA, G.; RECHT, L.; VARELA, A.
Revista:
JOURNAL OF GEOMETRY AND PHYSICS
Editorial:
ELSEVIER SCIENCE BV
Referencias:
Lugar: Amsterdam; Año: 2014 vol. 86 p. 241 - 241
ISSN:
0393-0440
Resumen:
Left invariant metrics induced by the p-norms of the trace in the matrix algebra are studied on the general linear group. By means of the Euler?Lagrange equations, existence and uniqueness of extremal paths for the length functional are established, and regularity properties of these extremal paths are obtained. Minimizing paths in the group are shown to have a velocity with constant singular values and multiplicity. In several special cases, these geodesic paths are computed explicitly. In particular the Riemannian geodesics, corresponding to the case p = 2, are characterized as the product of two one-parameter groups. It is also shown that geodesics are one-parameter groups if and only if the initial velocity is a normal matrix. These results are further extended to the context of compact operators with p-summable spectrum, where a differential equation for the spectral projections of the velocity vector of an extremal path is obtained.