QR decomposition

QR decomposition factors a matrix A into A = Q*R, where Q has orthonormal columns (an orthogonal matrix: its columns are unit-length and mutually perpendicular) and R is upper-triangular. Think of it as the Gram-Schmidt process packaged neatly: Q holds the cleaned-up perpendicular directions, and R bookkeeps how the original columns were built from them.

Its claim to fame is least-squares fitting, the math behind fitting a line or curve to noisy data. Because Q's columns are orthonormal, they do not amplify rounding errors, which makes solving the least-squares problem far more numerically stable than forming and inverting other matrices directly.

In short, when you want a dependable answer to an over-determined problem (more equations than unknowns, no exact solution), QR is the standard tool engineers and statisticians reach for.