Inverse of sum of rank one matrices: How to reuse the parts of inverse if there is a change in one of the rank one matrix?

Question

Let us suppose $$A := \left( \underbrace{I + \sum_{k=1}^K \alpha_{k} u_ku_k^T}_{:=M} \right)^{-1},$$ where $\alpha_k \in \mathbb{R}$, $u_k \in \mathbb{R}^n$, and $I$ is an invertible matrix.

We can find the inverse of matrix $M$ using the method proposed by K. S. Miller or see example here.

Question:

Now, if $\alpha_k$ has changed/updated, then how to reuse matrix $A$ as efficiently as possible without recomputing the inverse of $M$ from scratch.

The reason I am asking is that there is an iterative algorithm where $\alpha_k$ keeps updating within iterations.

Answer 1

Suppose that the updated matrix is

$$ B = \left(I + \sum_{k=1}^K \beta_{k} u_ku_k^T \right)^{-1}. $$ We can compute the updated inverse using the Woodbury matrix identity if we note that $B = A + UCV$ with $U$ the matrix with columns $u_1,\dots,u_K$, $V = U^T$, and $C = \operatorname{diag}(\beta_1 - \alpha_1, \beta_2 - \alpha_2, \dots, \beta_K - \alpha_K)$.