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Let $A\in \mathbb{R}^{\mathbb{N}\times \mathbb{N}}$ be a banded matrix ($A_{ij}=0$ for all $|i-j|>K$) with banded inverse (probably different bandwidth $\tilde K$). The question is, if principal submatrices $B_n := A|_{\{1,\ldots,n\}\times \{1,\ldots,n\}}$ also have banded inverses with a uniform bound for the bandwidths $\tilde K_n$?

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  • $\begingroup$ It looks like the nullity theorem could help here. $\endgroup$ Mar 9, 2017 at 6:42
  • $\begingroup$ I don't think it would help. If you take the statement and forget about $A^{-1}$ being banded, then it is definitely false. $\endgroup$ Mar 9, 2017 at 23:39
  • $\begingroup$ Sorry, maybe the comment gave the wrong impression -- I don't think it's an immediate consequence of this theorem; but it might be useful in attacking the problem. Just suggesting a possible line. $\endgroup$ Mar 10, 2017 at 6:33

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