QRdecomposition

QR decomposition, also known as QR factorization, is a matrix factorization in which a given matrix A is expressed as a product A = Q R. Here Q is an orthogonal (real) or unitary (complex) matrix, and R is an upper triangular matrix. For real matrices, orthogonality means Q^T Q = I and, for complex matrices, Q^H Q = I.

If A is m × n with m ≥ n and full column rank, a reduced QR decomposition exists: A = Q̃ R, where Q̃ is m × n with orthonormal columns and R is n × n upper triangular. In the square case with full rank, Q is m × m orthogonal and R is m × m upper triangular; in numerical work the reduced form is most common.

Common algorithms to compute QR decompositions include the Gram–Schmidt process (classical and modified for numerical stability), Householder reflections, and Givens rotations. Householder and Givens methods are preferred in practice due to improved numerical stability and efficiency, especially for large or sparse matrices.

Applications of QR decomposition are widespread. It is used to solve linear least squares problems by reducing the system Ax ≈ b to Rx = Q^T b, which yields x = R^{-1} Q^T b for the reduced form. It also underpins the QR algorithm, an iterative method for computing eigenvalues of square matrices, and serves as a stable alternative to LU decomposition in many numerical linear algebra tasks.

See also: Gram–Schmidt process, Householder reflection, Givens rotation, QR algorithm.