Question

Define Householder matrix.

Answer 1

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In linear algebra, a Householder transformation is a linear transformation that describes a reflection about a plane or hyperplane containing the origin. The Householder transformation was introduced in 1958 by Alston Scott Householder. Its analogue over general inner product spaces is the Householder operator.

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Answer 2

The reflection hyperplane can be defined by a unit vector {\textstyle v} (a vector with length {\textstyle 1}) which is orthogonal to the hyperplane. The reflection of a point {\textstyle x} about this hyperplane is the linear transformation:

{\displaystyle x-2\langle x,v\rangle v=x-2v\left(v^{\textsf {H}}x\right),}

where {\textstyle vis given as a column unit vector with Hermitian transpose {\textstyle v^{\textsf {H}}}.