I just want to make sure I'm thinking about this correctly. I've been given a matrix A and I need to find the eigenvalues and eigenvectors geometrically. I have found the eigenvalues. It wasn't too hard. Now I have graphed everything and I need to find the eigenvectors. This is where I get really confused. I've spent over an hour reading other posts and examples on other websites, but I'm still super confused. I still gave the problem a shot and I've posted a picture of my work. Can someone just confirm that this is correct? It is the last problem on the page, so it starts on the sixth line from the bottom of the page and continues onto the next picture. In the next picture those are the eigenvectors I got.
Question: Find the eigenvalue and eigenvector of A geometrically. In this problem
$$A=
\begin{bmatrix}
0 & 1 \\
1 & 0 \\
\end{bmatrix}
$$
and it is a reflection in the line y=x.
Best Answer
So, you found that the possible eigenvalues are (a) $1$ and (b)$-1$.
Geometrically this means that in the direction associated to each of these values the transformation (a) leaves the vector as it is and (b) multiplies it by $-1$, which means it reflects it through the origin.
For (a) ask yourself which direction is unchanged when you reflect it through the line $x=y$. It shouldn't be hard to see that any vector of the form $(x,x)$ is unaffected by this transformation.
For (b) ask yourself which direction is reflected through the origin by your transformation. In this case you can see that the vectors of the form $(y,-y)$ are what you're looking for.
The you have that $E_1=Span\{(1,1)\}$ and $E_{-1} = Span\{(1,-1)\}$.
PS. It's actually A LOT easier to do the math and then try to give it a geometrical meaning. So when doing this kind of problems try first to find the eigenvectors with the usual method and then see what does the transformation do to them.