Moore--Penrose Inverse, Parabolic Subgroups, and Jordan Pairs

E. Tevelev

Abstract: 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 {\it 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.

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