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Let $A\in M_n(K)$ be a square matrix over a field $K$. The notion of inverse matrix was generalized by Moore and Penrose for real and complex matrices (also called pseudo-inverse $A^{\dagger}$ of $A$, satisfying $AA^{\dagger}A=A, A^{\dagger}AA^{\dagger}=A^{\dagger}$ and $AA^{\dagger}$ Hermitian). This was again generalized to arbitrary fields with involutory automorphism of $K$ by Pearl, but the existence depends on additional rank conditions.

Question: What is known on the existence of a generalized inverse $A'$, which just satisfies $AA'A=A$, over an arbitrary field ? Do we also need an additional assumption, perhaps on the characteristic of $K$, or again some rank condition ?

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up vote 3 down vote accepted

The condition $AA'A=A$ means that the semigroup of matrices is regular. This is really true: see Clifford - Preston, The Algebraic Theory of Semigroups, sec.2.2, ex. 6(g).

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Thank you, Boris. Do you have some other reference where this is proved ? (I have no access to the book). –  Dietrich Burde Jun 26 '13 at 17:11
    
@ Dietrich Burde: Have you access to Semigroups and Combinatorial Applications by Gérard Lallement? If yes, then see in it Proposition 4.2. –  Boris Novikov Jun 26 '13 at 17:44
    
Yes, thank you. And the proof is really not difficult. –  Dietrich Burde Jun 28 '13 at 20:27
    
@ Dietrich Burde: OK. Good luck! –  Boris Novikov Jun 28 '13 at 20:30

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