#### Householder Reflections

For completeness, this section derives the Householder reflection matrix from geometric considerations [451]. Let denote the*projection matrix*which orthogonally projects vectors onto ,

*i.e.*,

*difference vector*between and , its orthogonal projection onto , since

*minus*this difference vector gives a

*reflection*of the vector about :

*reflecting*about --a so-called

*Householder reflection*.

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