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Householder matrix

  1. May 4, 2010 #1
    We show that the Householder matrix [tex]H=I-2ww^{*}[/tex] is unitary.

    Given a vector [tex]x \in \textbf{C}^{n}[/tex] and an integer k with 1<k<n, we derive a formula for a Householder matrix with the property that [tex](Hx)_{i}=0[/tex] for i>k and [tex](Hx)_{i}= x_{i}[/tex] for i<k. Make sure to choose the signs so that the formula is numerically stable.

    I can prove that the householder matrix is unitary.

    Now, with the properties of [tex](Hx)_{i}[/tex] described above, I think that the Householder matrix will be like a unit matrix, except that for the last rows (for i>k), we only have zeroes.

    According to the formula of the Householder matrix, [tex]H=I-2ww^{*}[/tex] , we have that [tex]2ww^{*}[/tex] must be a matrix with only zeroes in the rows (i<k), and with 1's on the diagonals of the last k rows and 0 elsewhere.

    But I am not sure how the vectors w will look like.
  2. jcsd
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