Find upper Hessenberg by Householder transformation

Question

I have a matrix that looks like this:

$$ \begin{pmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \epsilon & 0 & 0 & 0 \\ \end{pmatrix} $$

Compute a Householder transformation so that the matrix becomes an Upper Hessenberg.

Is there any special property of this matrix that I can use (since there is a matrix element that is a variable)?

I tried to find vector $u$ but it did not work out.

score 1 · Accepted Answer · answered Nov 29 '13 at 02:20

1

$$ Q=I-\frac{2}{2\epsilon^2} \begin{pmatrix}0\\-\epsilon\\0\\\epsilon\end{pmatrix} \begin{pmatrix}0\\-\epsilon\\0\\\epsilon\end{pmatrix}^T = \begin{pmatrix} 1&0&0&0\\ 0&0&0&1\\ 0&0&1&0\\ 0&1&0&0 \end{pmatrix}\;,\quad QA=\begin{pmatrix} \color{red}0&\color{red}1&\color{red}0&\color{red}0\\ \color{red}\epsilon&\color{red}0&\color{red}0&\color{red}0\\ 0&\color{red}0&\color{red}0&\color{red}1\\ 0&0&\color{red}1&\color{red}0 \end{pmatrix}\;. $$

answered Nov 29 '13 at 02:20

Algebraic Pavel

22,928

from where did you get the vector $v$? – dresden Nov 29 '13 at 09:37
You want to transform $x=[0,0,0,\epsilon]^T$ to $y=[0,\epsilon,0,0]^T$, hence $u$ in $Q=I-2uu^T$ is given by $u=(x-y)/|x-y|_2=[0,-\epsilon,0,\epsilon]^T/(2\epsilon)^{1/2}$. – Algebraic Pavel Nov 29 '13 at 16:37
thanks, that really helped! – dresden Nov 29 '13 at 17:08

Find upper Hessenberg by Householder transformation

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