QR decomposition

QR decomposition factors a matrix as the product of an orthogonal matrix and an upper-triangular matrix. It supports stable solutions of linear systems, least-squares problems, and eigenvalue computations.

Gram–Schmidt process: An algorithm that orthogonalizes a sequence of vectors by subtracting their projections onto previously constructed vectors. It constructs the orthogonal factor by turning the matrix’s columns into an orthonormal basis.
Linear least squares: The problem of finding a vector that minimizes the Euclidean norm of the residual in an overdetermined linear system. QR solves it by transforming the system and solving a triangular subproblem.
Orthogonal complement: The subspace of vectors orthogonal to every vector in a given subspace. It supplies additional orthonormal directions when a reduced factor must be extended to a full basis.
Singular value decomposition: A matrix factorization into two orthogonal factors and a diagonal matrix of nonnegative singular values. Unlike QR, it reveals singular values and provides a direct account of numerical rank.
Householder transformation: An orthogonal reflection that maps a vector onto a chosen direction, often a coordinate axis. Householder reflections zero entries below the diagonal to form the triangular factor.
QR algorithm: An iterative method for computing eigenvalues by repeatedly applying QR factorizations to a matrix. It repeatedly factors and recombines matrices to drive them toward triangular form.
Matrix rank: The dimension of a matrix’s column space, equal to the number of independent rows or columns. Rank determines how many independent directions the orthogonal factor must represent.
Cholesky decomposition: A factorization of a symmetric positive-definite matrix into a triangular factor and its transpose. It is often cheaper when normal equations produce a positive-definite matrix, but can be less stable for least squares.
Givens rotation: A rotation acting in a coordinate plane, leaving all other coordinates unchanged. A sequence of these rotations can eliminate selected entries while preserving sparsity.
Linear system: A set of linear equations that can be written as a matrix equation. For square systems, factoring the coefficient matrix can make repeated solves efficient.

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