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group inverse (Definition)

Let $ A$ be an $ n \times n$ matrix over $ \mathbb{R}$. A group inverse for $ A$ is an $ n \times n$ matrix $ X$ such that

$\displaystyle AXA$ $\displaystyle = A$ (1)
$\displaystyle XAX$ $\displaystyle = X$ (2)
$\displaystyle AX$ $\displaystyle = XA.$ (3)

Such a matrix, when it exists, is unique and is denoted by $ A^{\char93 }$. A group inverse is a special case of a Drazin inverse.



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Cross-references: Drazin inverse, matrix
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This is version 2 of group inverse, born on 2007-04-30, modified 2007-04-30.
Object id is 9306, canonical name is GroupInverse.
Accessed 866 times total.

Classification:
AMS MSC: 15A09 (Linear and multilinear algebra; matrix theory :: Matrix inversion, generalized inverses)
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