Artículos de revistas
The left invariant metric in the general linear group
Fecha
2014-12Registro en:
Andruchow, Esteban; Larotonda, Gabriel Andrés; Recht, Lázaro; Varela, Alejandro; The left invariant metric in the general linear group; Elsevier Science; Journal Of Geometry And Physics; 86; 12-2014; 241-257
0393-0440
Autor
Andruchow, Esteban
Larotonda, Gabriel Andrés
Recht, Lázaro
Varela, Alejandro
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.