I have been reading a theorem related with the existence of the outer generalized inverse of a matrix where i have certain difficulties to understand the theorem.
Theorem is as follows.
Let $A\in\mathbb{C}^{m\times n}$, rank$(A) = r$, and let $T$ and $S$ be a subspace of $\mathbb{C^n}$ and $\mathbb{C^m}$, respectively, with $\dim (T )= \dim (S^\perp)=t\leq r$.
Then, $A$ has a $\{2\}$ - inverse X such that $R(X) = T$ and $N(X) = S$ iff one of the following condition is satisfied (where $R(X)$ and $N(X)$ denots the range and null space of $X$, respectively)
$AT\oplus S$ = $C^{m}$
$P_{S}{^\perp} AT = S^{\perp}$
$A^*S^\perp\oplus T^\perp$ = $C^{n}$
$P_T~ A^*S^\perp = T$
$\{2\}$ - inverse of a matrix $A$ is a $n\times m$ matrix $X$ satisfying matrix equation $XAX = X$.
All above conditions are equivalent.
$P_{L.M}$ stands for the projection on to the space $L$ parallel to $M$ while $P_{L}$ stands for orthogonal projection onto sub space $L$ parallel to $L^\perp$.
Earlier i have posted same theorem where i was not clear about $AT$. Now that is cleared to me by answer given by David mitra .
I need a proper interpretation of these terms $P_{S}{^\perp} AT = S^{\perp}$, $P_T~ A^*S^\perp = T$, $A^*S^\perp$. it is given in the theorem that $AT\oplus S$ = $C^{m}$ that means that there must exist projction operator $P_{AT}{S}$ .E projection onto subspace $AT$ parallel to $S$. Also we have $dim (AT) = \dim S^\perp$.
I don't need proof. It has some connection with direct sum of sub spaces and projection associated with that.
I just need their interpretation. I have to use this theorem for my own work. But how can i use if the things are not cleared to me? I really need help so that i can proceed further.
Heartily thanks for giving me your precious time.