Householder transformation

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.

The Householder operator may be defined over any finite-dimensional inner product space ${\displaystyle V}$ with inner product ${\displaystyle \langle \cdot ,\cdot \rangle }$ and unit vector ${\displaystyle u\in V}$ as

It is also common to choose a non-unit vector ${\displaystyle q\in V}$, and normalize it directly in the Householder operator's expression:

Such an operator is linear and self-adjoint.

If ${\displaystyle V=\mathbb {C} ^{n}}$, note that the reflection hyperplane can be defined by its normal vector, a unit vector ${\textstyle {\vec {v}}\in V}$ (a vector with length ${\textstyle 1}$) that is orthogonal to the hyperplane. The reflection of a point ${\textstyle x}$ about this hyperplane is the Householder transformation:

where ${\displaystyle {\vec {x}}}$ is the vector from the origin to the point ${\displaystyle x}$, and ${\textstyle {\vec {v}}^{*}}$ is the conjugate transpose of ${\textstyle {\vec {v}}}$.

The matrix constructed from this transformation can be expressed in terms of an outer product as:

is known as the Householder matrix, where ${\textstyle I}$ is the identity matrix.