Suppose we have a complex matrix $\mathbf{M}$. Let $\mathbf{M}^+=(\mathbf{M}^\*\mathbf{M})^{-1}\mathbf{M}^\*$ be the pseudo-inverse of $\mathbf{M}$, where $^\*$ denotes the conjugate transpose. Let $\mathbf{G}=(a\mathbf{H}+\mathbf{M})^+$, i.e. $\mathbf{G}$ is the pseudo-inverse of $a\mathbf{H}+\mathbf{M}$, for some matrix $\mathbf{H}$ and scalar $a$.


If we assume that $a \ll 1$, how I can show that $\mathbf{G}$ can be approximated by the Taylor expansion as $$\mathbf{G}\approx \mathbf{M}^+ (\mathbf{I}-a\mathbf{H}\mathbf{M}^+), $$ where $\mathbf{I}$ is the identity matrix. 



Approximate $\mathbf{G}=(a\mathbf{H}+\mathbf{M})^+$ by Taylor expansion
to test if $G=(aH+M)^+=M^+(I-aHM^+ + {\cal O}(a^2))$ you can multiply from the right with $aH+M$ (using $M^+M=I$) and see if you obtain the identity to first order in $a$:


$$M^+(I-aHM^+ + {\cal O}(a^2))(aH+M)=M^+(aH+M-aHM^+M+{\cal O}(a^2))=I+{\cal O}(a^2)$$


