QR Decomposition Using Householder Reflections

QR decomposition is a fundamental matrix factorization technique in linear algebra, frequently implemented using Householder reflections. This algorithm decomposes any matrix A into the product of an orthogonal matrix Q and an upper triangular matrix R. In practical implementations, Householder reflections create zeros below the diagonal elements by applying successive orthogonal transformations. Each reflection acts on a column vector to eliminate its components below the pivot position, effectively transforming it into a multiple of a standard basis vector. The algorithm proceeds column-wise: for each column k, we compute the Householder vector v that reflects the column's subdiagonal elements to zero. The transformation matrix H = I - 2vvᵀ/vᵀv is then applied to the remaining submatrix. This process builds Q as the product of all Householder matrices and R as the final upper triangular matrix. QR decomposition with Householder reflections offers superior numerical stability compared to alternatives like Gram-Schmidt orthogonalization. It serves as the backbone for critical numerical algorithms including linear least squares solutions (via A\b in MATLAB), eigenvalue computations (QR algorithm), and singular value decomposition. The method finds extensive applications in computer vision for camera calibration, signal processing for filter design, and data science for principal component analysis. Key implementation considerations include: handling the sign choice to avoid cancellation errors, efficient storage of Q as a product of Householder vectors rather than explicit matrix formation, and optimized BLAS level-2 operations for matrix-vector products. Modern libraries like LAPACK implement this through routines such as DGEQRF for double precision matrices.