I would like to know if exist the analytical solution for the eingenvalue to the following tridiagonal matrix:
$A_{i,i}=i$ for $i=1,2,...,N$,
$A_{i,i+1}=A_{i+1,i}=1$
$$ \left(\begin{matrix} 1 & 1 & 0 &0 &...&0&0\\ 1 & 2 & 1 & 0&...&0&0\\ 0 & 1 & 3 &1&...&0&0\\ 0&0&1&4&1&...&0&\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots \end{matrix}\right) $$
For simplicity, elements can assume small values of type: $A_{i,i+1}=A_{i+1,i}=\epsilon$ or $A_{i,i}=1+i\epsilon$ .
I solved the homogeneous case with a recurrence relation, but in this case I have a complicated recurrence relation. If anyone has an idea how I can work with the eigenvalues and eigenvectors of this matrix, I would be very grateful.