2008-11-26

Laplacian matrix and its eigenvalues

Japanese version

Today, I read a paper, ``A signal processing approach to fair surface design'' (G. Taubin), but I did not understand it. It said that the eigenvalues of a Laplacian matrix like Equation 1 is Equation 2.
Eq 1Eq 2

Today, my friend CR taught me why this is. Actually, it is not so difficult. Here, we can back to the basics of eigenvalues, a characteristics function:
Also we know a Laplacian matrix is a second order differentiation. Therefore, it is
One of the solution of this differential equation is exponential function (since the second derivative is back to the original one except a constant factor). If you remember the Euler's equation, you can get Equation 2.

Additionally, Equation 1 is symmetric. Therefore, its eigenvalues are real and eigenvectors are orthogonal. Moreover, this is a diffusion equation and you can see the relationship with Fourier
basis.

Now, I can see why the paper's title is ``signal processing approach.'' But, one who is not familiar with mathematics like me, it is hard to understand the logic as ``The Equation 1's eigenvalues can be written as Equation 2. Therefore, we could decompose the signal...''

Of course, this is easy for the professionals and it is too verbose to write this like this article. However, at least I need this kind of explanations. Thanks CR.

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