Geodesic
Moore-Penrose Inverse, Parabolic Subgroups, and Jordan Pairs
Journal of Lie Theory, Tome 12 (2002) no. 2, pp. 461-481

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A Moore-Penrose inverse of an arbitrary complex matrix A is defined as a unique matrix A+ such that AA+A = A, A+AA+ = A+, and AA+, A+A are Hermite matrices. We show that this definition has a natural generalization in the context of shortly graded simple Lie algebras corresponding to parabolic subgroups with "aura" (abelian unipotent radical) in simple complex Lie groups, or equivalently in the context of simple complex Jordan pairs. We give further generalizations and applications.
Classification : 15A09, 17B45, 22E10
Mots-clés : Lie groups, generalized inverse, Moore-Penrose inverse, simple Lie algebras, bilinear forms, Moore-Penrose orbits, parabolic subgroups 
E. Tevelev. Moore-Penrose Inverse, Parabolic Subgroups, and Jordan Pairs. Journal of Lie Theory, Tome 12 (2002) no. 2, pp. 461-481. http://geodesic.mathdoc.fr/item/JOLT_2002_12_2_a8/
@article{JOLT_2002_12_2_a8,
     author = {E. Tevelev},
     title = {Moore-Penrose {Inverse,} {Parabolic} {Subgroups,} and {Jordan} {Pairs}},
     journal = {Journal of Lie Theory},
     pages = {461--481},
     year = {2002},
     volume = {12},
     number = {2},
     zbl = {1002.15008},
     url = {http://geodesic.mathdoc.fr/item/JOLT_2002_12_2_a8/}
}
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