Compute or approximate the inverse of $\mathbf{A} + \mathbf{u}\mathbf{v} + \lambda \mathbf{I}$

Question

The Sherman–Morrison formula states that $$\left(\mathbf{A} + \mathbf{u}\mathbf{v}^T\right)^{-1} = \mathbf{A}^{-1} - \frac{\mathbf{A}^{-1}\mathbf{u}\mathbf{v}^T\mathbf{A}^{-1}}{1 + \mathbf{v}^T \mathbf{A}^{-1} \mathbf{u} }.$$ My question is that $\mathbf{A}$, $\mathbf{u}$ and $\mathbf{v}$ have same assumption with Sherman–Morrison formula, and how can I compute or approximate the inverse $$\left(\mathbf{A} + \mathbf{u}\mathbf{v}^T + \lambda \mathbf{I}\right)^{-1},$$ where $\lambda > 0$ is very small constant.

Answer 1

Given $B$ any invertible matrix with $\mu\ne 0$ minimum eigenvalue, then $1/\mu$ is the spectral radius of $B^{-1}$, so, if $\lambda<\mu$, we have

$$ (B+\lambda I)^{-1} = B^{-1}\sum_{i=0}^{\infty} B^{-i}\lambda^i(-1)^i $$

so you can approximate the inverse with a partial sum of the series, with $B=A+uv^T$.