Given a matrix $A \in \mathbb{R}^{n\times m}$, and its perturbation 
$$
A\_p = A + \Delta
$$
is there a way to represent 
$$
(A\_p)^{\star}= (A)^{\star} + f(\Delta)
$$
where $(A\_p)^{\star}$ ($(A)^{\star}$) is the pseudo-inverse of $A\_p$ ($A$)?
What can be said about the spectral norm of $f(\Delta)$?



Perturbations on the pseudoinverse of a matrix
substituting $A\_p=A+\Delta$ into the definition $A\_p^\star=\lim\_{\epsilon\rightarrow 0}(A\_p^\ast A\_p+\epsilon I)^{-1}A\_p^\ast$, and linearizing in the perturbation $\Delta$, gives $A\_p^\star=A^\star+f(\Delta)$ with


$$f(\Delta)= -A^\star\Delta A^\star+
\lim\_{\epsilon\rightarrow 0}(A^\ast A+\epsilon I)^{-1}\Delta^\*P+{\rm order}(\Delta^2),$$


with $P=I-AA^\star$ the orthogonal projector onto the range of $A$. If $\Delta^\ast P=0$ the simple result


$$A\_p^\star=A^\star-A^\star\Delta A^\star+{\rm order}(\Delta^2)$$


is obtained.


