For infinite matrices, Jaffard's theorem states that if $(A(k,l))_{k,l\in \mathbb{Z}}$ is invertible and satisfies $$ A(k,l) \leq C (1+\left|k-l\right|)^{-r}, $$ for some $C>0$, then $$ A^{-1}(k,l) \leq C' (1+\left|k-l\right|)^{-r}, $$ for all $k,l\in \mathbb{Z}$ and for some $C'>0$. That is if the initial matrix exhibits polynomial off-diagonal decay, so does its inverse.

I am wondering if there is a complementary version for finite matrices. Note that for finite matrices such a result would be interesting if $C'$ can be bounded in terms of $C$ and $r$.

This paper claims that $C'$ depends only on $r$, $C$, and some parameter $b$ such that $$1>b\geq \left\|I-A\right\|_2.$$ However, there seem to be some critical typos in the surrounding discussion, and they refer (cf. p.5 of the preprint linked) to Jaffard's paper for a proof that $C'$ depends only on certain constants (see loc. cit. for which). The latter is in French, so I couldn't verify this.

So I have three questions:

- Does any one have a pointer to a version of Jaffard's theorem for finite matrices?
- Or alternatively, is there a clear argument to bound $C'$ in terms of $C$, $r$, and possibly $b$?
- Supposing that there is indeed such a bound, can it be generalized to settings where $\left\|I-A\right\|_2 \geq 1$?

Propriétés des matrices «bien localisées» près de leur diagonale et quelques applications. 1990 ", which is referenced in the reference in this OP. $\endgroup$ – Peter Heinig Oct 5 '17 at 8:25