3
$\begingroup$

Prof told us that $\textrm{refl}_n(p) = p - 2\textrm{proj}_n(p)$, where $p$ is a $d$ dimensional vector. After going over the geometrical definition, this made sense.

However, this got me stuck on this: If $\Omega$ is hyperplane in $d$ dimension, that passes through $0$ vector with unit normal $\mathbf n = [n_1, n_2, n_3, ... , n_d]$, how can we denote matrix $[R_\Omega]$ expressed in terms of $\mathbf n$? (where $R_\Omega$ is reflection through $\Omega$).

If $\mathbf n$ was not a unit normal, then it would make sense to apply the above equation of $\mathrm{refl}_n(p)$, but what happens in case of a unit normal? I understand what happens geometrically and algebraically in reflection, but need help building a matrix that represents such reflection.

$\endgroup$
2
$\begingroup$

If the projection matrix is $P$ then the reflection matrix is simply $R=I-2P$. So we need to find $P$.

The projection into the hyperplane with normal $\mathbf n$, is of the form $p-\alpha \mathbf n$. To verify the orthogonality to $\mathbf n$, $\alpha$ should be chosen as: $$ \alpha=<p,\mathbf n> $$ where $<.,.>$ is the inner product in the proper space.

To find the equivalent matrix representation of projection, we have to pick a basis $u_1,...,u_d$ which the matrix is representing the projection in this basis. A standard manipulation shows that: $$ P=I-\mathbf n\left[ \begin{matrix} <u_1,\mathbf n>,&\dots&,<u_d,\mathbf n> \end{matrix} \right]. $$

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.