Cfrgtkky

Definition: We say that a vector $xinmathbb{R}^n$ is monotone if, $x_1geq x_2 geq cdots geq x_n$ or $x_1 leq x_2 leq cdots leq x_n$.

Let $M=(m_{ij})_{1leq i,j leq n}$ be a matrix defined by $m_{ij}=1$ if $|i-j|leq frac{n}{2}$ and $m_{ij}=0$ otherwise. Show that the eigenvector corresponding to the second largest eigenvalue is monotone. (i.e., if $lambda_1geqlambda_2cdotsgeqlambda_n$ are the eigenvalues in non-increasing order and $v_1,v_2,ldots,v_n$ are the corresponding eigenvectors then $v_2$ is monotone).

We can prove this by finding a closed-form expression of the eigenvector, but I am trying to figure out if there is an other way, in fact one can show the same for the matrix defined by $m_{ij}=n-|i-j|$. Note that both are Toeplitz matrices, and Robinsonian matrices!

In this article the authors show that : if $M$ is a Robinsonian matrix and $L_M$ is the corresponding Laplacian, then the eigenvector corresponding to the second smallest eigenvalue (called the Fiedler vector) is monotone. I am wondering if there is a link?