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In linear algebra, a Householder transformation (also known as a Householder reflection or elementary reflector) is a linear transformation that describes a reflection about a plane or hyperplane containing the origin. The Householder transformation was used in a 1958 paper by Alston Scott Householder. [1]
A Householder reflection (or Householder transformation) is a transformation that takes a vector and reflects it about some plane or hyperplane. We can use this operation to calculate the QR factorization of an m-by-n matrix with m ≥ n. Q can be used to reflect a vector in such a way that all coordinates but one disappear.
For dense matrices, the left and right unitary matrices are obtained by a series of Householder reflections alternately applied from the left and right. This is known as Golub-Kahan bidiagonalization.
Together with a first step using Householder reflections and, if appropriate, QR decomposition, this forms the DGESVD [23] routine for the computation of the singular value decomposition. The same algorithm is implemented in the GNU Scientific Library (GSL).
"A block reflector is an orthogonal, symmetric matrix that reverses a subspace whose dimension may be greater than one." [1]It is built out of many elementary reflectors.. It is also referred to as a triangular factor, and is a triangular matrix and they are used in the Householder transformation.
This is a reflection in the hyperplane perpendicular to v (negating any vector component parallel to v). If v is a unit vector, then Q = I − 2vv T suffices. A Householder reflection is typically used to simultaneously zero the lower part of a column. Any orthogonal matrix of size n × n can be constructed as a product of at most n such ...
In linear algebra, the Householder operator is defined as follows. [1] Let be a finite-dimensional inner product space with inner product , and unit vector.Then : is defined by
This makes it twice as fast as algorithms based on QR decomposition, which costs about floating-point operations when Householder reflections are used. For this reason, LU decomposition is usually preferred.