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Let

$$X_i = \left(I - P\left(I - t_it_i^T\right)\right)^{-1}$$

where $P$ is an $N\times N$ matrix and $t_i$ is a vector of $N$ elements.

Is there a way to simplify this expression in order to calculate the inverse matrix only once for all $i$ values?

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I think I found a solution to this problem. As Federico Poloni also noted, we can use the Shelman-Morisson formula.

We can write $X_i=(I-P+Pt_it_i^T)^{-1}$

Let $A=I-P$, $u=Pt_i$ and $v=t_i$, then:

$X_i=(A+uv^{T})^{-1}$

We can use the Sheman-Morisson formulla:

$(A+uv^T)^{-1}=A^{-1}-\frac{A^{-1}uv^{T}A^{-1}}{1+v^{T}A^{-1}u}$

This way we can calculate $(I-P)^{-1}$(provided $I-P$ is invertible) only once for all values of $i$.

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If $\|P\|<1$ then $X_i = \sum_{k\ge 0} (P-Pt_i^{\top}t_i)^k$. This is analytic in $t_i$ and in $P$. Maybe this helps.

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