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I have a matrix of the form $A = bI - J$ where
1. $b$ is a large positive constant so that $A$ is positive definite
2. $J_{ij} = 0$ for $i=j$ and follows a power law off-diagnol. In index notation:

$$ A_{ij} = (b\delta_{ij} - \frac{J_0(1-\delta_{ij})}{|i-j|^\alpha}) \text{ where } \alpha \geq 1 $$

Is it possible to calculate $A^{-1}$ analytically? I hope to show that $A^{-1}$ also follows some type of power law as well as is suggested by numerics.

Any suggestions?

Tommi
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yakzo
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  • Of course it is possible by solving linear equations. Maybe you see some law if you try. But I'm afraid that's not what you want to know? – Horst Grünbusch Aug 28 '14 at 09:18
  • I want to know it for any finite dimension. – yakzo Aug 28 '14 at 09:29
  • @Travis Sorry, I must be misunderstanding. When you say "solving linear equations" do you mean Gauss-Jordan elimination? Doesn't this depend on having a specific matrix to perform the process on? – yakzo Aug 28 '14 at 11:57
  • @Travis Thanks! That last idea worked pretty well. – yakzo Sep 01 '14 at 08:37

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