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$\begingroup$ How can we show that if $G$ is generalized Inverse of $A$ ,
then $G^T$ is generalized Inverse of $A^T$
$\endgroup$ 1 Answer
$\begingroup$ We have G defined by the properties
$$\begin{align}
& AGA = A\\
& GAG = G \\
\end{align}$$
Taking the transpose of both sides of each equation
$$\begin{align}
& (AGA)^T = A^TG^TA^T = A^T\\
& (GAG)^T = G^TA^TG^T = G^T \\
\end{align}$$
gets you the property you want.
$\endgroup$ 
