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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:

  1. Does any one have a pointer to a version of Jaffard's theorem for finite matrices?
  2. Or alternatively, is there a clear argument to bound $C'$ in terms of $C$, $r$, and possibly $b$?
  3. Supposing that there is indeed such a bound, can it be generalized to settings where $\left\|I-A\right\|_2 \geq 1$?
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  • $\begingroup$ Just an 'offhand' thought: for question 1. and 2. it might be useful to try working with the Sherman–Morrison–Woodbury formula. Didn't try to answer the question proper along these lines. And a rule-of-thumb as to which direction in 'knowledge space' to look along: 'numerical linear algebra' and its many texts. Also: it might be useful to others to have a link to the journal version of " S. Jaffard: 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
  • $\begingroup$ no because in the infinite case the result says "there exists some C'..." I need a bound on C' in terms of the parameters b and C. $\endgroup$ – Ozzy Oct 8 '17 at 15:42
  • $\begingroup$ In the first paragraph, I think you want absolute value signs on the LHS of the inequalities, and also I think you need another hypothesis (else the diagonal matrix with entries getting arbitrarily small would be a counter-example). $\endgroup$ – Pace Nielsen Oct 13 '17 at 16:43

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