Suppose I have a block matrix as follows where $A \in \mathbb{R}^{n \times (n+p)}$ where $I_{n,n}$ is an $n \times n$ identity matrix and $B \in \mathbb{R}^{n \times p}$
$$A = [I \quad B]$$
I want to find the pseudo-inverse/generalized inverse of $A$. Is there a simple closed form for this? If necessary, perhaps assume that $rank(B) = p$.
Thanks!
It is easy to check that the rows are linearly independent (due to the $I$ block). Hence, the pseudoinverse is \begin{align*} A^{\dagger} &= A^T(AA^T)^{-1} \\ &= \begin{bmatrix}I \\ B^T\end{bmatrix}\left(\begin{bmatrix}I & B\end{bmatrix}\begin{bmatrix}I \\ B^T\end{bmatrix}\right)^{-1} \\ &= \begin{bmatrix}I \\ B^T\end{bmatrix}(I+BB^T)^{-1} \\ &= \begin{bmatrix}(I+BB^T)^{-1} \\ B^T(I+BB^T)^{-1}\end{bmatrix}. \end{align*}
You can check that $AA^{\dagger} = I$.