Suppose $A \in \mathbb{R}^p$ is the adjacency matrix of a weighted directed acyclic graph $D$ with vertex set $\left\{v\_{1}, v\_{2}, \ldots, v\_{p}\right\}$, i.e.
$$
 a\_{i j}=\left\{\begin{array}{lr}
w\left(v\_{i}, v\_{j}\right), & \text { if there is an arc from } v\_{i} \text { to } v\_{j} \\
0, & \text { otherwise }
\end{array}\right. 
$$
What I want to know is that for a specific $x \in \mathbb{R}^p$, does there exists a explicit display for
$$
 \big[ (I\_p - A)^{-1} x \big]\_{-j} - (I\_{p - 1} - A\_{-j, -j})^{-1} x\_{-j}
$$
where $x\_{-j} = (x\_1, \ldots, x\_{j - 1}, x\_{j + 1}, \ldots, x\_p)$, and $A\_{-j, -j}$ is the $A$ dropping $j$-th row and column.


PS: I have tried that


If $A$ is just the adjacency matrix, there exists $m \in \mathbb{N}$ such that
$$
 (I\_p - A)^{-1} = I + A + A^2 + \cdots + A^m
$$
and the $a\_{ij} \in \{ 0, 1\}$ will allow a simple display for $(I\_p - A)^{-1} x$. However, now we allow $a\_{ij} \in \mathbb{R}^+$. $A$ may not a nilpotent matrix. I do not know how to solve this problem.


Can anyone help me? Thanks in advance!



A property of directed acyclic graph
**Preliminary definition.** Let $\mathcal{S}$, $\mathcal{S}'$ be two complementary nonempty sets of indices, i.e., $\mathcal{S}\cup \mathcal{S}'=\left\{1,2,\ldots,p\right\}$ and $\mathcal{S}\cap \mathcal{S}'=\emptyset$.


Define $\mathcal{E}\_{\mathcal{S}}\overset{\Delta}=\left[\left(I-A\right)^{-1}x\right]\_{\mathcal{S}}-\left(I\_{\mathcal{S}}-A\_{\mathcal{S}}\right)^{-1}x\_{\mathcal{S}}$, where $\left[y\right]\_{\mathcal{S}}$ is the sub-vector indexed by $\mathcal{S}$. Remark that this conforms to a generalized form for the *error term* in your question where you assume $\mathcal{S}=\left\{1,2,\ldots,j-1,j+1,\ldots,p\right\}$ and $\mathcal{S}'=\left\{j\right\}$.


**Preliminary remark.** I will refer to $\mathcal{E}\_{\mathcal{S}}$ as the *error term* since it is the error committed by commuting the projection $\left[\cdot\right]\_{\mathcal{S}}$ with the underlying operations (product and matrix-inversion). In a sense, the question can be rephrased as: Does the error of commuting the projection $\left[\cdot\right]\_{-j}$ with the underlying operations admit an *amenable closed-form* expression in terms of $A$ and $x$?


For the first term of $\mathcal{E}\_{\mathcal{S}}$, observe that $\left[\left(I-A\right)^{-1}x\right]\_{\mathcal{S}}=\left[x+Ax+A^2x+\ldots\right]\_{\mathcal{S}}=\sum\_{i=0}^{\infty} \left[A^ix\right]\_{\mathcal{S}}=\sum\_{i=0}^{\infty} \left[A^i\right]\_{\mathcal{S}}x\_{\mathcal{S}}+\left[A^i\right]\_{\mathcal{S}\mathcal{S}'}x\_{\mathcal{S}'}$.


For the second term of $\mathcal{E}\_{\mathcal{S}}$, we have


$$\left(I\_{\mathcal{S}}-A\_{\mathcal{S}}\right)^{-1}x\_{\mathcal{S}}=\sum\_{i=0}^{\infty} \left(A\_{\mathcal{S}}\right)^ix\_{\mathcal{S}}.$$


Therefore, via combining the two terms, we have


$$\mathcal{E}\_{\mathcal{S}}=\sum\_{i=1}^{\infty} \left(\left[A^i\right]\_{\mathcal{S}}-\left(A\_{\mathcal{S}}\right)^i\right)x\_{\mathcal{S}}+\sum\_{i=0}^{\infty}\left[A^i\right]\_{\mathcal{S}\mathcal{S}'}x\_{\mathcal{S}'},$$


which yields


$$\mathcal{E}\_{\mathcal{S}}=\left(\left[(I-A)^{-1}\right]\_{\mathcal{S}}-(I\_{\mathcal{S}}-A\_{\mathcal{S}})^{-1}\right)x\_{\mathcal{S}}+\left[(I-A)^{-1}\right]\_{\mathcal{S}\mathcal{S}'}x\_{\mathcal{S}'}.$$


In your particular case, where $\mathcal{S}=\left\{1,2,\ldots,j-1,j+1,\ldots,p\right\}$ and $\mathcal{S}'=\left\{j\right\}$, we can further simplify the above expression via the following inversion Lemma (e.g., Matrix Analysis, Horn):


$$\left(B\_{\mathcal{S}}\right)^{-1}=\left[B^{-1}\right]\_{\mathcal{S}}-\left[B^{-1}\right]\_{\mathcal{S}\mathcal{S}'}\left(\left[B^{-1}\right]\_{\mathcal{S}'}\right)^{-1}\left[B^{-1}\right]\_{\mathcal{S}'\mathcal{S}} \,\,\,(\star).$$


Set $B:=\left(I-A\right)$ in the inversion Lemma $(\star)$, then


$$\mathcal{E}\_{\mathcal{S}}=\frac{1}{\left(\left[\left(I-A\right)^{-1}\right]\_{jj}\right)}\left(\left[\left(I-A\right)^{-1}\right]\_{\mathcal{S}\mathcal{S}'}\left[\left(I-A\right)^{-1}\right]\_{\mathcal{S}'\mathcal{S}}\right)x\_{\mathcal{S}}+\left[(I-A)^{-1}\right]\_{\mathcal{S}\mathcal{S}'}x\_{j},$$


which under your notation goes by


$$\mathcal{E}\_{\mathcal{S}}=\frac{1}{\left(\left[\left(I-A\right)^{-1}\right]\_{jj}\right)}\left(\left[\left(I-A\right)^{-1}\right]\_{-j,j}\left[\left(I-A\right)^{-1}\right]\_{j,-j}\right)x\_{-j}+\left[(I-A)^{-1}\right]\_{-j,j}\,x\_{j}.$$


**Final remark.** Interestingly, the error term $\mathcal{E}\_{\mathcal{S}}$ does not depend on the entries of the $(p-1)\times (p-1)$ sub-matrix $\left[(I-A)^{-1}\right]\_{-j,-j}$ -- which is the *bulk* of the $p\times p$ matrix $\left(I-A\right)^{-1}$.


