Editing Householder transformation (section)

====Example====

consider the normalization of a vector of 1's 

<math>\vec v=\frac{1}{\sqrt{2}}\begin{bmatrix} 1\\1 \end{bmatrix}</math>

Then the Householder matrix corresponding to this vector is

<math>P_v=\begin{bmatrix}1&0\\0&1\end{bmatrix}-2(\frac{1}{\sqrt{2}}\begin{bmatrix} 1\\1 \end{bmatrix})(\frac{1}{\sqrt{2}}\begin{bmatrix} 1&1 \end{bmatrix})</math>

<math>=\begin{bmatrix}1&0\\0&1\end{bmatrix}-\begin{bmatrix} 1\\1 \end{bmatrix}\begin{bmatrix} 1&1 \end{bmatrix}</math>

<math>=\begin{bmatrix}1&0\\0&1\end{bmatrix}-\begin{bmatrix}1&1\\1&1\end{bmatrix}</math>

<math>=\begin{bmatrix}0&-1\\-1&0\end{bmatrix}</math>

Note that if we have a vector representing a coordinate in the 2D plane

<math>\begin{bmatrix}x\\y\end{bmatrix}</math>

Then in this case <math>P_v</math> flips and negates the x and y coordinates, in other words

<math>P_v\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}-y\\-x\end{bmatrix}</math>

Which corresponds to reflecting the vector across the line <math>y=-x</math>, which our original vector <math>v</math> is normal to.