# Searching for a right pseudoinverse

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:

1. 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.

2. 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?

I think you mean $P^{+R} = R \pmatrix{A^{-1}\cr 0\cr}$, so $P P^{+R} = (A |B) R ^{-1} R \pmatrix{A^{-1}\cr 0\cr} = I$.
More generally, this will work for any $n \times n$ invertible matrix $R$ such that the first $m$ columns of $P R$ have rank $m$.