While I was playing around with some matrices I had the need to expand the matrix inverse of . While doing so I came across two different tools, one known as the Woodbury identity and the other known as the Neumann series expansion. While messing with the aforementioned results I found out that there is an pretty cute and trivial way to prove the Neumann series expansion using the Woodbury inequality in a recursive way.
Theorem (Woodbury inequality)
Let’s consider , and then we could write:
We could easily prove this result just showing that the direct product of yield the identity matrix.
I’ll now introduce what is called the Neumann series expansion, with this term we identify the following result: . Is possible to prove that the series mentioned above converges, and therefore the above identity make sense, if operator norm of is less or equal to . To prove what we mentioned above we usually use the following argument:
We first define and then we could easily prove that:
Cutey Proof For Neumann Series Expansion
Let’s decompose using the Arnoldi iterations , once we write , then we could easily write:
We then expand using the Woodbury identity again:
substituting this in the above equation we obtain:
therefore we have, , once we apply recursively the Woodbury identity to expand we obtain: .
This seemed a nice thing for my first post, therefore this is all from me today !