Find the eigenvalues and eigenvectors of $A$ geometrically. $$A=
\begin{bmatrix}-1 & 0 \\
       0 & 1 
\end{bmatrix}
$$
Reflection in the $y$-axis. 
I can solve this WITHOUT the reflection meaning I can find the eigenvalue and eigenvector of this. However, I don't know what exactly I should be doing with the reflection. Do I just reflect the eigenvector or do I reflect matrix $A$? 
EDIT for the comments:
Do I reflect the eigenvector? Let's solve this problem: 
\begin{bmatrix}-1- λ & 0 \\
       0 & 1 - λ
\end{bmatrix}
λ = -1 or 1.
Therefore: Eigenvalue = -1,1 and Eigenvector using -1 will be:
\begin{bmatrix}1 \\
       0
\end{bmatrix}
Do I reflect this eigenvector? Is this what reflection over the Y-axis means? Or Do I only reflect A after plugging in the Eigenvalue for λ?
 A: Observe that
$
\begin{bmatrix}-1 & 0 \\
       0 & 1 
\end{bmatrix}
\begin{bmatrix}1\\
       0 
\end{bmatrix}=-1
\begin{bmatrix}1\\
       0 
\end{bmatrix}
$
and
$
\begin{bmatrix}-1 & 0 \\
       0 & 1 
\end{bmatrix}
\begin{bmatrix}0\\
       1 
\end{bmatrix}=1
\begin{bmatrix}0\\
       1 
\end{bmatrix}
$.
A: Geometrically, what does it mean for a vector $v$ to be an eigenvector of $A$? We know that $Av = \lambda v $ for some $\lambda \in \mathbb{R}$, but that means for any $t \in\mathbb{R}$ we have that $Atv = tAv = t\lambda v$, and so we have $A tv \in \mathbb{R} v$ where $\mathbb{R} v$ is the set of all multiples of $v$. This means that as a subspace, $A$ takes $\mathbb{R}v$ to $\mathbb{R} v$, or $A(\mathbb{R} v) = \mathbb{R} v$.  Now geometrically, what linear subspaces does a reflection leave invariant?
A: The matrix,
\begin{equation}
 A = \begin{bmatrix}-1 & 0 \\
       0 & 1 
\end{bmatrix},
\end{equation}
is a reflection across the y-axis. When you multiply a vector by $A$, it gives you the same vector, but reflected across the y-axis.
Its eigenvectors are
$v_1 = 
\begin{bmatrix}1\\
       0 
\end{bmatrix}$
and
$v_2=
\begin{bmatrix}0\\
       1 
\end{bmatrix}$ with corresponding eigenvalues $\lambda_1 = -1$ and $\lambda_2 = 1$.
There is no reflection necessary if all they are asking for is the eigenvalues and eigenvectors, as they are not asking you to reflect anything.
