I have been studying the SVD algorithm recently and I can understand how it might be used for compression but I am trying to figure out if there is a perspective of SVD where it can be seen as a low pass filter.

So, before we discuss SVD, I want to check if my understanding of eigenvalue decomposition is correct.

So, the eigenvalue decomposition of a square matrix can be written as:

$$A = VDV^{-1}$$

Where $V$ is the matrix where each column corresponds to an eigenvector of $A$ and $D$ is the diagonal matrix where the diagonal entry corresponds to the corresponding eigenvector. So, I can perform compression using eigenvalue decomposition by setting the eigenvalues under some threshold to 0. In this case, the smaller eigenvalues will have a relatively shrinking effect on the rows of $V^{-1}$ and will overall contribute less. I am hoping this reasoning is correct?

The main question that I have and this relates to eigenvalue decomposition as well as SVD is whether there is some relationship to the frequency content of the signal. Do the smaller eigenvalues contribute to high-frequency components of the signal? So is the compression algorithm acting like a low pass filter and depending on the threshold set is essentially stopping the high frequency signals to pass through and basically acting as a smoothing oiperator or is there no relationship there?


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