I am looking to express the pseudo-inverse of a matrix,
$\mathbf{B} = \mathbf{A} + \mathbf{E}$
where $\mathbf{E}$ is a small random perturbation, as a series expansion. For a real and square matrix $\mathbf{B}$, I am aware that the Taylor expansion (to first order),
$\mathbf{B^{-1}} \approx \mathbf{A^{-1}}-\mathbf{A^{-1}}\mathbf{E}\mathbf{A^{-1}} $
is valid. However, I am dealing with a complex non-square matrix $\mathbf{B}$. Is the above expansion also valid for this case?
Links to any relevant texts would be greatly appreciated!
Since you have a guarantee that the matrix $B\in\mathbb{C}^{m\times n}$ is a full rank matrix (you are right, the rank cannot exceed $m$ and $n$), then the pseudoinverse of the matrix $B$ is given by
$$ (A^T A)^{-1} A^T $$ if $m\geq n$ and $$ A^T(AA^T)^{-1} $$ if $m < n$.
Hope that this helps, since you can now use the expansion for the inverse of the matrix $A^T A$ or $AA^T$.