# group inverse

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 $A^{\#}$. A group inverse is a special case of a Drazin inverse.

Title group inverse GroupInverse 2013-03-22 17:01:17 2013-03-22 17:01:17 Mathprof (13753) Mathprof (13753) 5 Mathprof (13753) Definition msc 15A09