Householder operator

In linear algebra, the Householder operator is defined as follows. Let $V$ be a finite-dimensional inner product space with inner product $⟨\cdot ,\cdot ⟩$ and unit vector $u\in V$. Then

$H_u:V\to V$

is defined by

$H_u(x)=x-2⟨x,u⟩u.$

This operator reflects the vector $x$ across a plane given by the normal vector $u$.

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

$H_q\left(x\right)=x-2\frac{⟨x,q⟩}{⟨q,q⟩}q.$

Properties

The Householder operator satisfies the following properties:

• It is linear; if $V$ is a vector space over a field $K$, then

$\mathrm{\forall }(\lambda ,\mu )\in K^2,\mathrm{\forall }(x,y)\in V^2,H_q(\lambda x+\mu y)=\lambda H_q\left(x\right)+\mu H_q\left(y\right).$

• It is self-adjoint.
• If $K=\mathbb{ℝ}$, then it is orthogonal; otherwise, if $K=\mathbb{ℂ}$, then it is unitary.

Special cases

Over a real or complex vector space, the Householder operator is also known as the Householder transformation.

References

• {{citation | last=Roman | first=Stephen