# Tridiagonal With Constant Off-Diagonals

Does anybody know about the eigenvalues of a tridiagonal matrix with constant off-diagonal elements? Such a matrix can be rescaled into

$$\left( \begin{array}{ccccccc} a_1 & 1 & 0& \ldots & & \\ 1 & a_2 &1 & \ldots & & \\ 0 & 1 & a_3 &\ldots &&\\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ & & & \ldots & a_{n-1} & 1 \\ & & & \ldots & 1 & a_n \end{array} \right)$$

where $a=(a_1, a_2, \ldots, a_n)$ is a real vector.

For the case where $a$ is proportional to the unit vector, this is done on MS in Eigenvalues of a certain tridiagonal matrix . How would you go about for the general case?

• It is a symmetric tridiagonal matrix. There are searchable results about that – Omnomnomnom May 17 '17 at 22:06
• Not just that, but the off-diagonals are constant. – Jennifer May 18 '17 at 6:53