Eigenvectors from complex eigenvalues

[zfolwick]

how does one systematically find the eigenvectors of a 2x2 (or higher) Real matrix given complex eigenvalues?

 

[vela]

The same way you do it when you have real eigenvalues.

 

[HallsofIvy]

For example, the eigenvalues of the matrix
$$\begin{bmatrix}0 & -1 \\ 1 & 0 \end{bmatrix}$$
are i and - i.

If < x, y> is an eigenvector corresponding to eigenvalue i then we must have
$$\begin{bmatrix}0 & -1 \\ 1 & 0 \end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}= \begin{bmatrix}- y \\ x\end{bmatrix}= \begin{bmatrix} ix \\ iy\end{bmatrix}$$

So we must have -y= ix and x= iy. Since 1/i= -i, those are equivalent. Any such eigenvector is of the form < x, y>= <iy, y>= y<i , 1>.