Householder Matrix Example

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Householder Transformations Example DEF is called Householder matrix

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Householder Transformations Example DEF is called Householder matrix

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Householder Transformations Example DEF is called Householder matrix

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Householder Transformations Example DEF is called Householder matrix

Householder matrix

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Householder matrix

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Householder matrix

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$\notag \begin{gathered} \left[\begin{array}{@{\mskip2mu}rr@{\mskip2mu}} 0 & -1 \\ -1 & 0 \end{array}\right], \quad \displaystyle\frac{1}{3} \left[\begin{array}{@{\mskip2mu}rrr@{\mskip2mu}} 1 & -2 & -2\\ -2 & 1 & -2\\ -2 & -2 & 1\\ \end{array}\right], \quad \displaystyle\frac{1}{2} \left[\begin{array}{@{\mskip2mu}rrrr@{\mskip2mu}} 1 & -1 & -1 & -1\\ -1 & 1 & -1 & -1\\ -1 & -1 & 1 & -1\\ -1 & -1 & -1 & 1\\ \end{array}\right], \\ \displaystyle\frac{1}{5} \left[\begin{array}{@{\mskip2mu}rrrrr@{\mskip2mu}} 3 & -2 & -2 & -2 & -2\\ -2 & 3 & -2 & -2 & -2\\ -2 & -2 & 3 & -2 & -2\\ -2 & -2 & -2 & 3 & -2\\ -2 & -2 & -2 & -2 & 3 \end{array}\right], \quad \displaystyle\frac{1}{3} \left[\begin{array}{@{\mskip2mu}rrrrrr@{\mskip2mu}} 2 & -1 & -1 & -1 & -1 & -1\\ -1 & 2 & -1 & -1 & -1 & -1\\ -1 & -1 & 2 & -1 & -1 & -1\\ -1 & -1 & -1 & 2 & -1 & -1\\ -1 & -1 & -1 & -1 & 2 & -1\\ -1 & -1 & -1 & -1 & -1 & 2 \end{array}\right]. \end{gathered}$

What Is a Householder Matrix? – Nick Higham

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$\notag \displaystyle\frac{1}{5} \left[\begin{array}{@{\mskip2mu}rrrr@{\mskip2mu}} -2 & -4 & -1 & 2\\ -4 & 2 & -2 & -1\\ -1 & -2 & 2 & -4\\ 2 & -1 & -4 & -2 \end{array}\right].$

What Is a Householder Matrix? – Nick Higham

$\notag X = \displaystyle\frac{1}{3} \left[\begin{array}{@{\mskip2mu}rrr} 2+\mathrm{i} & -1+\mathrm{i} & -1+\mathrm{i}\\ -1+\mathrm{i} & 2+\mathrm{i} & -1+\mathrm{i}\\ -1+\mathrm{i} & -1+\mathrm{i} & 2+\mathrm{i} \end{array}\right].$

What Is a Householder Matrix? – Nick Higham

ALAFF Householder transformation

$\pm \mathrm{i}$

What Is a Householder Matrix? – Nick Higham

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What Is a Householder Matrix? – Nick Higham

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$\notag P = \begin{bmatrix} I_n\\0 \end{bmatrix} + \alpha \begin{bmatrix} u\\v \end{bmatrix}u^T = \begin{bmatrix} I_n + \alpha u u^T\\ \alpha vu^T \end{bmatrix} \in \mathbb{R}^n.$

What Is a Householder Matrix? – Nick Higham

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$q_1$

What Is a Householder Matrix? – Nick Higham

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$n-r$

What Is a Householder Matrix? – Nick Higham

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$1/2$

What Is a Householder Matrix? – Nick Higham

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$(i,i)$

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Householder's Method

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Householder Matrix Example

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