I find it is hard to understand the application senario of the Matrix Inversion Lemma in non-special cases. Suppose I already computed $A^{-1}$ and want to find $\left(A+UCV \right)^{-1}$. The Matrix Inversion Lemma says
$\left(A+UCV \right)^{-1} = A^{-1} - A^{-1}U \left(C^{-1}+VA^{-1}U \right)^{-1} VA^{-1}$
where $A, U, C$ and V all denote matrices of the correct size. Specifically, $A$ is $n \times n$, $U$ is $n \times k$, $C$ is $k \times k$ and $V$ is $k \times n$.
What I don't understand is: instead of finding the inverse of $A+UCV$, now I need to find the inverse of $C^{-1}+VA^{-1}U$. I can see that when $k < n$, I'm looking for the inverse of a smaller matrix, which may saves me some computation. But in which other condition I can benefit from using this formula?
We have that \begin{align} &(A+UCV) \left(A^{-1}-A^{-1}U(C^{-1}+VA^{-1}U)^{-1}VA^{-1}\right)=\\ &=AA^{-1}-AA^{-1}U(C^{-1}+VA^{-1}U)^{-1}VA^{-1}+UCVA^{-1}-UCVA^{-1}U(C^{-1}+VA^{-1}U)^{-1}VA^{-1} \\ &=I-U(C^{-1}+VA^{-1}U)^{-1}VA^{-1}+UCVA^{-1}+U(C^{-1}+VA^{-1}U)^{-1}VA^{-1}-UCVA^{-1} \\ &=I \end{align} since \begin{align} -UCVA^{-1}U(C^{-1}+VA^{-1}U)^{-1}VA^{-1}&=-UC(-C^{-1}+C^{-1}+VA^{-1}U)(C^{-1}+VA^{-1}U)^{-1}VA^{-1} \\ &=U(C^{-1}+VA^{-1}U)^{-1}VA^{-1}--UCVA^{-1}. \end{align}
Note. The "philosophy" of this formula is that you already know the inverses of $A$ and $C$ and you want to express, using what you know, the inverse of $A+UCV$.