Decompositions

The directions a matrix merely stretches instead of rotating — the natural axes along which covariance structure, iterative processes, and optimization landscapes reveal themselves.

Rotate centered data onto its axes of maximal variance and keep the top few — the optimal linear compression, and the standard interview probe on eigenvectors, variance, and reconstruction error.

Every matrix — square or not — factors into rotate, stretch, rotate. The stretch factors rank what the matrix actually does, which is why SVD powers low-rank approximation, PCA, and stable least squares.