TITLE: Perturbations on the pseudoinverse of a matrix
QUESTION [5 upvotes]: 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 $f(\Delta)$?

REPLY [4 votes]: 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.