Hyperplane Reflection

Question

Given the reflection $R_H(−2, 2, 2, −3) = (−4, 0, −2, 1)$, what is the reflection of $R_H(−1, −1, −1, 3)$, and how I can find it ?

I tried to use $R_H = (2P_H b -I)b$, but I have no idea how to keep going on this exercise

thanks

Answer 1

Let

$$U=(−2, 2, 2, −3)^T \ \ \text{and} \ \ V=(−4, 0, −2, 1)^T \ \ \text{giving} \ \ W=U-V=(2, 2, 4, -4)^T$$

(recall: the $^T$ operators mean that $U,V,W$ are transposed of line vectors, therefore are column vectors).

Let us consider the hyperplane $P$ with unit normal vector

$$N=W/\|W\|=\frac{1}{\sqrt{40}}(2, 2, 4, -4)^T$$

Taking the formula of the Householder reflection matrix (similar to yours) with respect to hyperplane $P$:

$$H=I-2NN^T=\left[\begin{array}{rrrr}0.8&-0.2&-0.4& \ \ 0.4\\ -0.2&0.8&-0.4&0.4\\ -0.4&-0.4&0.2&0.8\\ 0.4&0.4&0.8&0.2\\ \end{array}\right]$$

It remains to apply this matrix to vector

$$(-1, -1, -1, \ 3)^T$$