Koszul algebra

In Linear algebra, define the Householder operator as follows.

Let $V$ be a finite dimensional inner product space with unit vector $u\in V$ Then, the Householder operator is an operator $H_u:V\to V$ defined by

$H_u(x)=x-2⟨x,u⟩u$

where $⟨\cdot ,\cdot ⟩$ is the inner product over $V$. This operator reflects the vector $x$ across a plane given by the normal vector $u$.

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

The Householder operator has numerous properties such as linearity, being self-adjoint, and is a unitary or orthogonal operator on V.

References

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