I have a question regarding the Woodbury matrix identity. In general, the Woodbury matrix identity is as follows
$(\mathbf{A} + \mathbf{u} \mathbf{u}^T)^{-1} = \mathbf{A}^{-1} - \frac{\mathbf{A}^{-1}\mathbf{u} \mathbf{u}^T\mathbf{A}^{-1}}{1 + \mathbf{u}^T \mathbf{A}^{-1}\mathbf{u} }$ where $\mathbf{u}$ is a $n \times 1$ vector and $\mathbf{A}$ is a $n \times n$ matrix.
Is there any formula to calculate the sum of this expression? i.e., $(\mathbf{I} + \sum_{k = 1}^{K}a_k \mathbf{u}_k \mathbf{u}_k^T)^{-1}$ where $\mathbf{I}$ is an identity matrix of size $n \times n$?
Thank you very much in advance