How to Derive the Sherman-Morrison Base Formula Before diving into the Sherman-Morrison formula, Meyer in Matrix Analysis and Applied Linear Algebra, pg. 124, starts with
$$
(I+cd^T)^{-1} = I - \frac{cd^T}{1+dc^T}
$$
where $c,d$ are vectors, and says "it's straightforward to verify by direct multiplication".
I took that to mean multiply right (in essence, both) sides by $(I+cd^T)$, and see if I could reach the identity matrix on RHS, so
$$
I + cd^T - \frac{cd^T (I + cd^T)}{1+d^Tc}
$$
$$
= I + cd^T - \frac{I cd^T (1+cd^T)}{1+d^Tc}
$$
$$
= I + cd^T - cd^T = I
$$
It seemed to work. 
I am wondering though how one could derive this equality by just starting from $(I+cd^T)^{-1}$. How did the mathematician go about finding that?
 A: Well, $I + \alpha cd^T$ with $\alpha \in \mathbb{R}$ is the general form of a matrix that fixes vectors orthogonal to $d$ by multiplication on the left and fixes vectors orthogonal to $c^T$ by multiplication on the right. If $I + cd^T$ has an inverse, then its inverse also fixes these subspaces by multiplication and so is in this form. Now, one can solve for $\alpha$.
$$(I + cd^T) (I + \alpha cd^T) = I + cd^T + \alpha cd^T + \alpha cd^Tcd^T \\
= I + (1 + \alpha + \alpha d^T c) cd^T$$
For this to be identity, $1 + \alpha + \alpha d^T c$ must be zero, so $$\alpha = \frac{-1}{1 + d^Tc}$$
A: The important concept here is that you can "use" matrices as arguments in analytic functions (provided that eigenvalues of these matrices lie in the domain of analyticity). To further simplify, if you have a function representable by a power series, you can plug a square matrix instead of a complex variable.
Consider $$f(x) = \frac{1}{1+x} = \sum_{k\ge 0} (-1)^k x^k,$$ defined on $\{|x|<1\}$.
Now replace $x$ by $cd^T$ (under the hypothesis that all eigenvalues of $cd^T$ lie within a unit disk):
$$(I+cd^T)^{-1} = \sum_{k\ge 0} (-1)^k (cd^T)^k = I - \sum_{k\ge 1}(-1)^{k-1}(cd^T)^k = I - c\left(\sum_{k\ge 1}(-1)^{k-1}(d^Tc)^{k-1}\right)d^T = I - c(I+(d^Tc))^{-1}d^T .$$
You need to deal with the case when the eigenvalues lie both inside the disk and outside the disk, but the idea remains the same.
A starting point in wiki: https://en.wikipedia.org/wiki/Matrix_function#Cauchy_integral
