Singular Value Decomposition

Every matrix — square or not, invertible or not — splits into three clean pieces: a rotation, a stretch along perpendicular axes, and another rotation. That's the Singular Value Decomposition, and it's arguably the most useful theorem in applied mathematics. It compresses images, recommends movies, powers latent semantic search, and underlies PCA. When you 'truncate' the smallest stretch factors, you keep the essence and discard the noise.