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In linear algebra, the Householder operator is defined as follows. Let V {\displaystyle V\,} be a finite dimensional inner product space with inner product ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } and unit vector u ∈ V {\displaystyle u\in V}. Then
is defined by
This operator reflects the vector x {\displaystyle x} across a plane given by the normal vector u {\displaystyle u}.
It is also common to choose a non-unit vector q ∈ V {\displaystyle q\in V} , and normalize it directly in the Householder operator's expression
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