# Moore-Penrose Pseudoinverse

Consider an $m\times n$ matrix over a field $K$, $A\in M(m, n, K)$. The Moore-Penrose pseudoinverse of $A$ is a matrix $A^+\in M(n, m, K)$ such that,

$AA^+A=A$

$A^+AA^+=A^+$

$(AA^+)^*=AA^+$

$(A^+A)^*=A^+A$

For $x\in M(m, 1, K)$, let $A|x$ denote the matrix obtained by appending $x$ after the last column of $A$. Given two column matrices $u,v\in M(m, 1, K)$, is there a clear relationship between $(A|u)^+$ and $(A|v)^+$?

I realize the notion of "clear relationship" is vague, but any sort of algebraic identity relating to two, or evidence of no such identity would be extremely useful. I have not found literature in either case. Thank you!

Going from $(A\mid u)$ to $(A\mid v)$ is a special case of a "rank-1 update" which is expressed generally as a relation between a matrix $A$ and $(A + cd^T)$. There is a formula for the rank-1 update of the Moore-Penrose which says that $(A + cd^T)^+ = A^+ + G$ where $G$ is a certain matrix too complicated to write here. It can be found (with proper references) in Section 3.2.7 of the "Matrix Cookbook" by Kaare Brandt Petersen and Michael Syskind Pedersen.
This will work for matrices from $\mathbb{R}$. I am not so sure about other funny/fancy fields.