Q:

Eigenvalues and Corresponding Eigenspace Bases

Could someone describe the eigenvalues of $ \left( \begin{array}{cc}
2 & 1   \\
-1 & 2 
 \end{array} \right) $, as well as the bases of the corresponding eigenspaces? 
I received eigenvalues of $ \lambda = 2+i, 2-i$. And $ E_{2+i} = \operatorname{span}\{[1,i]^T\}$ and $E_{2-i} = \operatorname{span}\{ [1,-i]^T\}$. It also wants me to illustrate the action of $A$ on eigenvectors. How would I do so?  
Are these solutions correct?

A:

You are correct except the eigenvectors are
$$
E_{2+i} = (1,i)^t\\
E_{2-i} = (1,-i)^t
$$
The action of $A$ on eigenvalue $E_\lambda$ is to multiply it by $\lambda$

