# Householder matrix

1. May 4, 2010

### math8

We show that the Householder matrix $$H=I-2ww^{*}$$ is unitary.

Given a vector $$x \in \textbf{C}^{n}$$ and an integer k with 1<k<n, we derive a formula for a Householder matrix with the property that $$(Hx)_{i}=0$$ for i>k and $$(Hx)_{i}= x_{i}$$ 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 $$(Hx)_{i}$$ 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, $$H=I-2ww^{*}$$ , we have that $$2ww^{*}$$ 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.

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