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I was wondering what we can say about the eigenvectors of a matrix $A$ fullfilling $Ax =0$ where $A$ is symmetric with a diagonal equal to one and every row sums up to 0. Obviously this is a generalisation of the graph Laplacian with positiv weights. It s easy to see that the constant vector is an Eigenvector to 0. I m not sure if there are more. Any help or pointers would be greatly appreciated.

Edit: the underlying graph is connected

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  • $\begingroup$ If you take the graph Laplacian for a disconnected graph, say $\begin{pmatrix}1&-1&0&0\\-1&1&0&0\\0&0&1&-1\\0&0&-1&1\end{pmatrix}$, then there are clearly more... $\endgroup$
    – Mateusz Kwaśnicki
    Commented Sep 17, 2021 at 9:50
  • $\begingroup$ True ! I should have mentioned that the grap also has to be connected. $\endgroup$
    – RZA Chris
    Commented Sep 17, 2021 at 10:22
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    $\begingroup$ Then how about $\begin{pmatrix}1&-1&1&-1\\-1&1&-1&1\\1&-1&1&-1\\-1&1&-1&1\end{pmatrix}$? :-) $\endgroup$
    – Mateusz Kwaśnicki
    Commented Sep 17, 2021 at 10:36
  • $\begingroup$ Haha you r killing me ;) thanks a lot $\endgroup$
    – RZA Chris
    Commented Sep 17, 2021 at 12:43

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