# Eigenvalues of Symmetric Pseudo-Toeplitz Matrix

I'm interested in calculating the eigenvalues of a symmetric tridiagonal matrix where the left and right diagonals are 1 and the main diagonal has entries $a_i \in \mathbb{R}$. I was wondering if there was a reference where the eigenvalues are derived.

\begin{equation*} \left[ \begin{array}{ccccccc} a_1 & 1 & 0 & 0 & 0 & \cdots & 0 \\ 1 & a_2 & 1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & a_3 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & a_4 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & 1 & a_{n-1} & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & a_n \end{array} \right] \end{equation*}

The special case that I am particularly interested in, is when $a_2 = a_3 = \cdots = a_{n-1} = 0$ and $a_1,a_n \neq 0$. Any help would be greatly appreciated.

## migrated from mathoverflow.netFeb 25 '16 at 14:11

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