So how does PCA actually work? (karthikvedula.com)

Principal Component Analysis (PCA) is a foundational technique in AI and machine learning used to reduce high-dimensional data into simpler, more manageable forms while preserving the most critical information. This article demystifies PCA by breaking down its mechanics through the example of projecting 2D data onto a 1D line, illustrating how maximizing the variance of projected points retains the most information. Key prerequisites for understanding include linear algebra concepts such as eigenvectors and eigenvalues, as well as basic calculus, specifically optimization methods like Lagrange multipliers. The core insight lies in expressing the variance of projected data as a quadratic form involving the covariance matrix and the direction vector of projection. PCA seeks the direction (principal component) that maximizes this variance under a unit length constraint, leading to an eigenvalue problem where the principal components correspond to the covariance matrix’s eigenvectors and the variance explained corresponds to eigenvalues. This connection highlights why PCA efficiently captures dominant patterns in data, enabling dimensionality reduction, visualization, and improved model performance by focusing on the most significant variance directions. For AI/ML practitioners, grasping PCA’s underlying math empowers more informed application and interpretation of this ubiquitous technique.