Eigenvectors and eigenvalues of nonsymmetric Tridiagonal matrix Hi, the question is following: We have one matrix
$$\begin{pmatrix}
  -\beta & \Delta & 0 & 0 &\cdots & 0 & 0 & 0 \newline
  \beta & -(\beta+\Delta) & \Delta & 0 &\cdots & 0 & 0 & 0 \newline
  0 & \beta & -(\beta + \Delta) & \Delta &\cdots & 0 & 0 & 0 \newline
  \vdots  & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots& \vdots \newline
  0 & 0 & 0 & 0 &\cdots & \beta & -(\beta + \Delta) & \Delta \newline
  0 & 0 & 0 & 0 & \cdots & 0 & \beta & -(\beta + \Delta)
 \end{pmatrix}.
$$
Is it possible to find analytically all eigenvalues and eigenvectors of this matrix? What I found here is quite similar, but not exactly the same. The element in the last row and last column $-(\beta +\Delta)$ can be replaced by $-\Delta$ if it simplifies the solution. $\beta>0$, $\Delta>0$.
Thanks
 A: Various commenters have pointed out that one can easily get asymptotic estimates for the eigenvalues as $n \rightarrow \infty$ (specifically, with $\beta, \Delta > 0$ they converge to $-2 (\beta + \Delta) \cos(\pi k/2 n)^2$ for $k = 1$ to $n$).  OTOH,  you specify in the comments that you require the eigenvalues in exact form. This is not possible. Even for $n = 5$, $\beta = 2$, and $\Delta = 1$, the characteristic polynomial of the matrix above is
$$x^5 + 14 x^4 + 70 x^3 + 150 x^2 + 129 x + 32$$
whose splitting field has Galois group $S_5$,  and hence there cannot be any exact formula (in radicals) for the roots by a theorem of Abel. This suggests that it is highly unlikely that there exists any "exact" formula for the roots, although "exact formula" is a somewhat nebulous notion, since of course one can  define  a function $f_{n,k}(\beta,\Delta)$ to be the $k$th smallest root of the corresponding characteristic polynomial.
A: There are such formulas, but beware: they are rather intimidating.
Look at this paper:
"EXPLICIT EIGENVALUES AND INVERSES OF TRIDIAGONAL TOEPLITZ MATRICES WITH FOUR PERTURBED CORNERS" by WEN-CHYUAN YUEH and SUI SUN CHENG2, ANZIAM J. 49(2008), 361–387.
