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Principal Components

  1. Given data matrix $X$ with $N$ rows (samples) and $k$ columns (features) – assume each feature has mean zero.
  2. The matrix $Q=\frac{1}{N}X^{\intercal}X$ is symmetric and its entries are the variances/covariances.
  3. If $v$ is a vector, then $Xv$ is called a “score” – a synthetic measure of the data.
  4. The variance of the score is $v^{\intercal}Qv$.
  5. Critical points of variance are eigenvectors of $Q$.
  6. These critical directions are called “principal components”.