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In many applications, an SVD of a matrix is used to determine which features are important and which ones less important. For example, in image compression, the smallest singular values are often thrown away, making the data smaller without losing much information.

  • What is the relation between the eigenvalues of a matrix and the singular values?
  • Can data also be compressed by doing an eigendecomposition and throwing away the smallest eigenvalues?
  • Do further analyses on the compressed matrix give qualitatively similar results?

For example, imagine I have a large $10\text{M}$ by $10\text{M}$ matrix for which I know the eigenvalues. I want to perform an SVD, but I am only interested in the $20$ largest singular values. Computationally, an SVD on such a large matrix is expensive. Can I exploit my knowledge of the eigenvalues to compress the matrix to a small size, while keeping the largest singular values mostly unaffected?

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