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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.
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 ...
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.
A reflection about a line or plane that does not go through the origin is not a linear transformation — it is an affine transformation — as a 4×4 affine transformation matrix, it can be expressed as follows (assuming the normal is a unit vector): [′ ′ ′] = [] [] where = for some point on the plane, or equivalently, + + + =.
"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.
Reflections, like the Householder transformation. times a Hadamard matrix. In general, any operator in a Hilbert space that acts by permuting an orthonormal basis is unitary. In the finite dimensional case, such operators are the permutation matrices.