TITLE: Eigenvectors of a $2 \times 2$ matrix when the eigenvalues are not integers
QUESTION [4 upvotes]: How can I calculate the eigenvectors of the following matrix?
$$\begin{bmatrix}1& 3\\3& 2\end{bmatrix}$$
I calculated the eigenvalues. I got
$$\lambda_1 =  4.541381265149109$$
$$\lambda_2 = -1.5413812651491097$$
But, now I don't know how to get the eigenvectors. When I create a new matrix after I subtracted Lambda value from all the members of the matrix on the main diagonal and tried to solve the homogeneous system of equations, I get only null vector for both $\lambda_1$ and $\lambda_2$....
When I used this website for calculating eigenvalues and eigenvectors. I got these eigenvectors
$(0.6463748961301958, 0.7630199824727257)$ for $\lambda_1$ 
$(-0.7630199824727257, 0.6463748961301957)$ for $\lambda_2$
.... but have no idea how to calculate them by myself...
Is it even possible? ....or it's possible to calculate it numerically?

REPLY [0 votes]: Now I get it....
After I try to solve the system of 2 equations for Lambda 1.... I get:
v1 = 0.84712708838 * v2 
v1 = 0.84712708838 * v2 
So, the system has infinitely many solutions...
So, I can e.g. write that (0.84712708838 , 1) is the eigenvector for Lambda 1.
Similarly for Lambda 2.... it also has infinitely many solutions...
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