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I have good intuition regarding the eigenvectors and eigenvalues for a matrix describing a transformation in a Hilbert space and those of correlation or covariance matrices.

For Hilbert space transformations, the eigenvectors are the new unit vectors in which the transformation becomes just scaling, and the scaling factors are the eigenvalues.

For a correlation matrix, or a covariance matrix, it means the vectors which transform the data into independent axis, i.e. the coordinates in which the data no longer correlates, and the eigenvalues represent how much information the corresponding vector holds.

What's the intuition for the meaning of the eigenvectors and eigenvalues of an adjacency matrix?

Why does "dimensionality reduction" i.e. picking those vectors who's eigenvalues are the greatest and using them to represent the whole graph work on adjacency matrices?

mathreadler
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Mr.WorshipMe
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  • I don't really know about the adjacency matrix spectral theory. I know about it for the graph Laplacian. The intuition for the graph Laplacian is that if you imagine a substance diffusing on the edges of the graph (it taking the same time to go along any edge), this diffusion can be Fourier-decomposed into slower and faster modes. The slow modes correspond to "chokepoints". – Ian Aug 28 '17 at 13:54
  • Analogously to how we solve the 1D heat equation by Fourier decomposition, the eigenvalues of the graph Laplacian describe the speed of relaxation of these Fourier modes. The eigenvectors strictly speaking behave just as they do in the usual setting: you decompose your initial configuration into the eigenvectors and then relax the eigenvectors separately. Loosely speaking, the idea is that material flows from the large positive parts of the eigenvector to the large negative parts or vice versa (depending on the direction of the gradient of the initial condition, of course). – Ian Aug 28 '17 at 13:54

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