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.


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: $$ P=I-\mathbf n\left[ \begin{matrix} <u_1,\mathbf n>,&\dots&,<u_d,\mathbf n> \end{matrix} \right]. $$


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