Generalized Inverse of Transpose

Question

How can we show that if $G$ is generalized Inverse of $A$ ,

then $G^T$ is generalized Inverse of $A^T$

Answer 1

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.