Householder operator

In Linear algebra, define the Householder operator as follows.

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

where ${\displaystyle \langle \cdot ,\cdot \rangle }$ is the inner product over ${\displaystyle V\,}$. This operator reflects the vector ${\displaystyle x}$ across a plane given by the normal vector ${\displaystyle u}$.

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