Is there a simple analytical solution to obtain eigenvalues (and eigenvectors) for this type of tridiagonal matrices ? ( Off diagonal elements are identical and the matrix is symmetric)
$ \begin{pmatrix} a_{1}-k & k & 0 & 0 \\\ k & a_{2}-2*k & k & 0 \\\ 0 & k & -a_{2}-2*k & k \\\ 0 & 0 & k & -a_{1}-k \end{pmatrix} $
To obtain eigenvalues of symmetric tridiagonal matrices, I used this formula $P_{k} (λ)=(α_{k}−λ)P_{k-1}(λ)−β^2_{k-1}P_{k-2}(λ)$. But I could not find simplifications.
Thanks for answer.