# Monotone eigenvector of the normalized Laplacian

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 $$Lv=\lambda_2 v$$.