TITLE: How to find complex eigenvectors from complex eigenvalues?
QUESTION [6 upvotes]: I have this matrix that represents a $2 \times 2$ linear system and I am supposed to solve to find what $x(t)$ and $y(t)$ are. 
$\left[ {\begin{array}{cc}
   1 & 5 \\
   -1 & -3 \\
  \end{array} } \right] $
I got that my eigenvalues were $-1+i$ and $-1-i$. But how do I use these to find my eigenvectors?

REPLY [3 votes]: $$\left[ {\begin{array}{cc}
   1-(-1+i) & 5 \\
   -1 & -3-(-1+i) \\
  \end{array} } \right] =  $$
$$ \left[ {\begin{array}{cc}
   2-i & 5 \\
   -1 & -2-i \\
  \end{array} } \right]$$
One eigenvector is $$ \begin{pmatrix} 2+i\\-1\end{pmatrix}$$
and the other one is  $$ \begin{pmatrix} 2-i\\-1\end{pmatrix}$$