I got a positive-definite symmetric toeplitz matrix $\mathbf{A}$, for example, $\begin{equation} \mathbf{A} = \left[ \begin{array}{ccc} 1 &2 &3 \\ 2& 1& 2 \\ 3& 2& 1 \end{array} \right] \end{equation}$.
It can be eigen-decomposition as
$\mathbf{U}^T \mathbf{A}\mathbf{U} = \mathbf{\Lambda},$
where $\mathbf{U} = \left[ \begin{array}{cccc} \mathbf{u}_1 & \mathbf{u}_2 & \mathbf{u}_{3} \end{array} \right]$ is an orthogonal matrix, i.e.,
$\mathbf{U}^T \mathbf{U} = \mathbf{U} \mathbf{U}^T = \mathbf{I}$,
with $\mathbf{I}$ being the identity matrix, and
$\mathbf{\Lambda} = \mathrm{diag} \left( \lambda_1, \lambda_2, \lambda_{3} \right)$
is a diagonal matrix. The orthonormal vectors $\mathbf{u}_1, \mathbf{u}_2, \ldots, \mathbf{u}_{3}$ are the eigenvectors corresponding, respectively, to the eigenvalues $\lambda_1, \lambda_2, \lambda_{3}$ of the matrix $\mathbf{A}$, where these eigenvalues are ordered from the smallest to the largest, i.e., $0 < \lambda_1 \leq \lambda_2 \leq \lambda_{3}$.
Also, I have a special vector $\mathbf{d}$, whose each element has norm $1$, for example, $\mathbf{d} = \left[ \begin{array}{c} 1 \\ 1 \\1 \end{array} \right]$.
I found a particular situation, i.e., $\mathbf{d}^H\left(\mathbf{u}_1\mathbf{u}^T_3 + \mathbf{u}_3\mathbf{u}^T_1 \right) \mathbf{d} = 0$. ( the eigenvectors corresponding the smallest and largest eigenvalues, and $^H$ denotes complex-conjugate transpose.)
It seems that it can be generalized to general examples. But I wonder why?