Suppose, I have a matrix $P\in\mathbb{R}^{m\times n}$ with $\text{rank}(P)=m$ and I'm searching for a right pseudoinverse $P^{+R}$.
Since I'm working with symbolic matrices in a computer algebra system, simply taking a Moore-Penrose-inverse can lead to more complicated inverses. Instead I came up with the following idea:
I can always rearrange the columns of $P$ such that $\tilde{P}=PR=\left(A \big\vert B\right)$ with $A\in\mathbb{R}^{m\times m}$ and $\text{rank}(A)=m$, while $R$ is a permutation matrix.
The right pseudo inverse can then be calculated with $$ P^{+R}= R \begin{pmatrix} A^{-1}\\ 0 \end{pmatrix}. $$
This seems to solve my problem excellently. However, since this is no magic i was wondering if you know any references on this approach?