QRdekompositio

QRdekompositio, or QR-decomposition, is a matrix factorization that expresses an m-by-n matrix A as A = Q R, where Q is an orthogonal (or unitary in the complex case) matrix and R is an upper triangular (or upper trapezoidal) matrix. If n ≤ m, R is upper triangular; if n > m, R is upper trapezoidal. In practice, an economy-size version uses Q with fewer columns and a correspondingly smaller R.

This factorization provides an orthogonal basis for the column space of A and isolates the rank information in R. It is widely used to solve linear systems and least-squares problems: solving Ax ≈ b can be done by first computing y = Q^T b and then solving R x = y, which is straightforward for the triangular R. QR is also employed in numerical methods for eigenvalue problems and in various regression and projection techniques.

QR can be computed by several methods, each with different numerical properties. Gram–Schmidt provides a simple interpretation but suffers from loss of orthogonality in finite precision. More stable approaches use Householder reflections or Givens rotations. In many applications a pivoted QR (QR with column pivoting) is preferred, as it improves numerical stability and can be rank-revealing. Variants such as economy-size or reduced QR tailor the shapes of Q and R to the dimensions of A.

Key properties include Q^T Q = I (or Q^* Q = I in the complex case) and the fact that R is upper triangular. Diagonal entries of R can be constrained to be positive to ensure a unique factorization when pivoting is used.