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Making sense of principal component analysis, eigenvectors & eigenvalues

I am currently going through a PCA tutorial. However, I am a bit confused. For PCA, we calculate the covariance matrix and then find the eigen vectors and eigen values of the covariance matrix. These vectors are the principal components.

  • How are the eigenvectors and the principal components related?
  • Why does finding the covariance matrix from the data and its eigenvectors are the principal components?
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