Suppose I have a matrix $A_{n \times n} , n \ge 2$. Let $n_1,n_2,\cdots,n_k$ be natural numbers such that $\sum_{i=1}^k n_i = n.$ Define $n_i^* = n_1+\cdots+n_i$ Then , $A$ can be partitioned as :
$\begin{bmatrix} (A_i)_{n_i^* \times n_i^*} & U_i \\ V_i & W_i \end{bmatrix}$
where it is given that $\forall i\in \{1,\cdots,k-1\} : columnspace(U_i) \subseteq columnspace(A_i)$ and $rowspace(V_i) \subseteq rowspace(A_i)$
My goal is to recursively find the generalized inverse of matrix $A$ denoted as $G_A$ by defining a matrix $B^{(i)}$ such that :
$B^{(i)} = \begin{bmatrix} B^{(i)}_{1,1} & B^{(i)}_{1,2} \\ B^{(i)}_{2,1} & B^{(i)}_{2,2} \end{bmatrix}$ where $G_{A_i} = B^{(i)}_{1,1}$.
Eventually, $G_A = G_{A_k} = B^{(k)} = B^{(k)}_{1,1}$. Here's my working for the same.
I have been recursively able to find an expression for $B^{(i)}_{1,1}$ but have no clue how to find out a recursive expression for $B^{(i)}_{1,2}, B^{(i)}_{2,1}, B^{(i)}_{2,2}$
