How do eigenvectors relate to the phenomenon of overfitting in machine learning models?

Overfitting happens when a model learns noise instead of true patterns, often because it tries to explain every tiny variation in the training data. In linear models, the training data’s covariance matrix can be decomposed into eigenvectors and eigenvalues, which show the directions and magnitudes of variation. Directions with very small eigenvalues correspond to noise or irrelevant variation, and a model that gives them weight will fit that noise, leading to overfitting. For example, if a dataset has 100 features but only 5 truly informative ones, a model that projects onto all 100 eigenvectors will capture noise in the remaining 95 small‑variance directions. Regularization or dimensionality reduction (like keeping only eigenvectors with large eigenvalues) helps avoid this by ignoring those noisy directions, thus reducing overfitting.