Householderreflecties

Householderreflecties, named after Alston Scott Householder, are orthogonal (real case) or unitary (complex case) transformations that reflect vectors across a hyperplane in Euclidean space. In n-dimensional real space a Householder reflection can be written as H = I − 2 v v^T / (v^T v), where v is a nonzero vector. Equivalently, using a unit vector u, H = I − 2 u u^T. For any vector x, the transformed vector is Hx = x − 2 v (v^T x)/(v^T v). The hyperplane of reflection is the set of vectors x for which v^T x = 0.

Properties and interpretation: A Householder reflection is symmetric (H^T = H) and orthogonal (H^T H = I), and it is involutory (H^2 = I). Its eigenvalues are 1 with multiplicity n−1 and −1 along the direction of v, indicating that it leaves the reflection hyperplane fixed while flipping the opposite direction. In the complex setting, the commonly used form H = I − 2 w w^† requires w to be a unit vector and yields a Hermitian, unitary transformation.

Construction and applications: Householder reflections are widely used to simplify matrices while preserving numerical stability. In QR factorization, a reflection is chosen to zero out subdiagonal elements of a column. Repeatedly applying reflections produces an orthogonal matrix Q and an upper triangular matrix R, with Q being the product of the individual Householder reflections. Beyond QR, these reflections are employed in eigenvalue algorithms, solving linear systems, and various numerical linear algebra preconditioning techniques. They provide a robust, geometrically intuitive method for shaping matrix structures without introducing round-off errors that accumulate from more arbitrary operations.