Suppose that I have a sparse and invertible $n\times n$ real symmetric matrix $M$ such that $M$ has at most $k$ non-zero entries ($k<<n$) in any row or column. (The case I'm interested in is $k=6$).
One can show that for any matrix $M$ we have $$\|M^{-1}\|_2\leq \frac{\|M\|_2^{n-1}}{|\det(M)|}\tag{$*$}\label{*} $$
- Let $\alpha$ and $\beta$ be the absolute values of smallest and largest non-zero entries in $M$ respectively. Can I use \eqref{*} to get a bound on $\|M^{-1}\|_2$ in terms of $\alpha$ and $\beta$?
- Is \eqref{*} tight? Might there be another way to bound $\|M^{-1}\|_2?$
In fact, $M$ has more properties, but are a bit tedious to write down. In case the above information is not sufficient, please let me know in the comments so that I update the question.