$$A=
\begin{bmatrix}-1 & 0 \\
0 & 1
\end{bmatrix}
$$
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?
Best Answer
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} $.