We show that the Householder matrix [tex]H=I-2ww^{*}[/tex] is unitary.(adsbygoogle = window.adsbygoogle || []).push({});

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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Householder matrix

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**