I am trying to find the inverse of the following symmetric positive definite matrix
$$ \left(\begin{array}{6*c} 4& 1&0 & &\cdots&0\\ 1& 4& 1& &\huge0& \vdots \\ 0& \ddots& \ddots& \ddots& & \\ \vdots && \ddots& \ddots& \ddots&0\\ & \huge 0 & & 1& 4& 1\\ 0& \cdots& &0& 1& 4 \end{array}\right) $$
Of course the mathematical inverse of this matrix is full, however numerically I observed that the coefficients' magnitude of the inverse matrix are decreasing at an exponential rate around the diagonal. Far from the border effect (i.e. for $i$ sufficiently greater than $1$ and lower than $n$) the diagonal pattern is the same up to $10^{-20}$: ($\forall j, i \neq i', M^{-1}_{i'j} \approx M^{-1}_{ij} $). Moreover, I empirically check that this behavior does not depend on the dimension of the linear system $n$ (same pattern).Do you have any clue on how to justify this observation? Or better, an analytical form to compute this inverse?
Related
