In the context of machine learning, how do eigenvalues relate to the dot product of vectors in a feature space, particularly when considering dimensionality reduction techniques like PCA?

Eigenvalues in PCA measure how much variance a particular direction (eigenvector) captures in the feature space. When you project data onto an eigenvector, the dot product of each data point with that eigenvector gives a coordinate; the squared length of this coordinate, summed over all points, equals the eigenvalue times the number of points, showing the variance along that direction. Thus, larger eigenvalues correspond to directions where the dot products vary widely, indicating more informative features. For example, if a 2‑D dataset has covariance matrix [[4,2],[2,3]], its eigenvalues are 5 and 2, meaning the first principal component explains twice as much variance as the second. PCA keeps the components with the largest eigenvalues, reducing dimensionality while preserving most of the dot‑product‑derived variance.