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Denote $L=\left[\begin{array}{ccc} 1 & -1 & 0\\ -1 & 2 & -1\\ 0 & -1 & 1 \end{array}\right]$ and $M=\left[\begin{array}{ccc} 1\\ & 0\\ & & 0 \end{array}\right]$. It's well known that $L$ is the Laplacian matrix of a path of length 3.

Is there any theory about the relationship between the eigenvector of $L$ and $L+M$ ? Especially for the eigenvector corresponding to the second smallest eigenvalue of $L$ and that to the largest eigenvalue of $L+M$.

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Perturbating a Laplacian matrix in just one entry, and then attempting to analyse how that does alter its eigenvalues and eigenvectors, is pretty much an impossible task. However, there is some theory regarding your specific matrices. First, note that your matrix $L+M$ is actually the $(1,1)$-principal minor of the Laplacian matrix of a path of length $4$. (Denote it by $L_{P_4}$.) Then, you may use Cauchy's Eigenvalue Interlacing Theorem to relate their (sorted) eigenvalues: $$0 =\lambda_1(L_{P_4}) \leq \lambda_1(L+M) \leq \lambda_2(L_{P_4}) \leq \lambda_2(L+M) \leq \lambda_3(L_{P_4}) \leq \lambda_3(L+M) \leq \lambda_4(L_{P_4}).$$ Britanak, Yip and Rao (Discrete Cosine and Sine Transforms: General Properties, Fast Algorithms and Integer Approximations, $2006$) wrote that the columns of this matrix: $$\left[\sqrt{\dfrac{2}{n-0.5}}\,\sin\left(\dfrac{\pi j(k-0.5)}{n-0.5}\right)\right]_{j,k=1}^{n-1}$$ are eigenvectors of $L+M$ (where $n=4$), with corresponding eigenvalues $$\left[4\sin^2 \left(\dfrac{j-0.5}{n-0.5} \, \dfrac{\pi}{2}\right)\right]_{j=1}^{n-1}.$$ There may not be any theory that relates the eigenvectors and eigenvalues of $L$ with those of $L+M$ though, despite we know them analytically.

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