Linear Algebra - Reflection in a hyperplane 
We have a matrix $A$: $$ A = \dfrac{1}{7} \cdot 
 \begin{pmatrix}
  5 & -4 & -2 & 2 \\
  4 & -1 & -4 & 4 \\
  -2  & -4  & 5 & -2  \\
  2 & 4 & 2 & 5 
 \end{pmatrix} $$
The map $f_a : \mathbb{R}^4 \to \mathbb{R}^4 $ is a reflection in the hyperplane $H \subset \mathbb{R}^4 $. Determine $H$. 

I don't quite how to find the hyperplane $H$. I think the root of my problem is that I don't really understand what is meant mathematically with "reflection in a hyperplane". Can anyone clear up what is asked and a strategy to find the hyperplane?

A reflection about the hyperplane $H$ will fix vectors in $H$ and reflect other vectors of $\mathbb{R}^4$ across $H$.  Since vectors in $H$ are fixed by $f_A$, they are eigenvectors of $f_A$ with eigenvalue 1.  To find $H$, you must find the eigenspace of $f_A$ associated to the eigenvalue 1.
An example in a smaller dimension is the matrix
\begin{equation*}
A = \left(\begin{matrix}-1 & 0 \\ 0 & 1\end{matrix}\right),
\end{equation*}
which represents reflection in $\mathbb{R}^2$ about the $y$ axis.  The eigenvectors of this $A$ with eigenvalue 1 are vectors which lie on the $y$ axis.
