Maximum norm bound on a block diagonal matrix inverse

Given a block diagonal (sparse) symmetric matrix $S \in \mathbb{R}^{n\times n}$ and a low-rank symmetric matrix $L$, construct $R=S-L$. I believe that $R$ is, in general, weak block diagonally dominant, as $R_{ii}=S_{ii}-L_{ii}$ for diagonal blocks and $R_{ij} = -L_{ij}$ for $i\neq j$.

For the 2-block case, this would look like:

$R=\left[\begin{array}{cc} S_{11}-L_{11} & -L_{12}\\ -L_{21} & S_{22}-L_{22} \end{array}\right]$

I found a helpful source that if $R$ is strict block diagonally dominant, Ahlberg-Nilson-Varah bounds the max norm:

$\|R^{-1}\|_\infty\leq \frac{1}{\min_i \|R_{ii}^{-1}\|_\infty^{-1}-\sum_{j\neq i}\|L_{ij}\|_\infty}$.

What I'd like to know is if there is a similar bound for the off-diagonal blocks. That is for

$\|(R^{-1})_{ij}\|_\infty\forall i\neq j$

either in the weak or strict cases of block diagonally dominance.

• The $R = S-L$ need not be weakly diagonally dominant. For example take $S = I$ and $L = jj^T$, where $j$ is a vector with elements equal $1$. – Pawel Kowal Aug 29 '16 at 18:42
• you mean for the Ahlberg-Nilson-Varah bound to hold? – zzk Sep 2 '16 at 20:41