A generalized inverse of $A$ can mean a matrix $G$ with $AGA=A$; this condition alone usually does not make $G$ unique. Over $\mathbb R$ or $\mathbb C$, the Moore–Penrose inverse $A^+$ uniquely satisfies $AA^+A=A$, $A^+AA^+=A^+$, and Hermitian symmetry of $AA^+$ and $A^+A$. If $A=U\Sigma V^*$, invert its nonzero singular values to obtain $A^+=V\Sigma^+U^*$.
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