Let $u_0 \geq u_1 \geq \cdots \geq u_{n-1}$ be positive numbers and define a matrix $n\times n$ by $M_{i,j} = u_{\left|i-j\right|}$ for all $i,j$.

Let $L = I - D^{-1/2}MD^{-1/2}$ be the normalized Laplacian matrix, where $D$ is a diagonal matrix such that $D_{ii} = \sum_j M_{i,j}$ for all $i$. Let $\lambda_1 = 0 \leq \lambda_2 \leq \cdots \leq \lambda_n$ be the eigenvalues of $L$.

Show that there exists a vector $v\in\mathbb{R}^n$ such that $v_1 \geq v_2 \geq \cdots v_n$ and $Lv=\lambda_2 v$.


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