Householder (1953) first considered the matrix that now bears his name in the first couple of pages of his book. A Householder matrix for a real vector can be implemented in the Wolfram
Language as:
Trefethen and Bau (1997) gave an incorrect version of the formula for complex . D. Laurie gave a correct version
by interpreting reflection along a given direction not as
(1)
where
(2)
is the projection onto the hyperplane orthogonal to (since this is in general not a unitary transformation), but
as
(3)
Lehoucq (1996) independently gave an interpretation that still uses the formula , but choosing to be unitary.