Matrix solution for complex eigenvalues

Question

So I have the next matrix: $\begin{bmatrix}1 & -4\\2 & 5\end{bmatrix}$ for which I have to find the general solution of the system $X' = AX$ in each of the following situations. Also, find a fundamental matrix solution and, finally, find $e^{ tA}$, the principal matrix solution.

I have managed to determine the eigenvalues: $\lambda_1=3-2i, \ \lambda_2=3+2i$. But I got the eigenvectors to be null and I don't know what to do next. Can somebody help me, please?

Answer 1

Did you get your eigenvectors to be null because you forced them to be real? If you allow complex values, you should be able to find nonzero eigenvectors. Provided that the constants are chosen appropriately, the imaginary parts will cancel at the end leaving a real matrix.

Answer 2

You determined the eigenvalues correctly.

To find the eigenvectors for $\lambda_1=3-2i$, solve $\begin{bmatrix}1 & -4\\2 & 5\end{bmatrix}\begin{bmatrix}z_1\\z_2\end{bmatrix}=(3-2i)\begin{bmatrix}z_1\\z_2\end{bmatrix}$.

I.e., $\begin{bmatrix}z_1-4z_2\\2z_1+5z_2\end{bmatrix}=\begin{bmatrix}(3-2i)z_1\\(3-2i)z_2\end{bmatrix}$.

I.e., $z_1(-2+2i)-4z_2=0$ and $2z_1+(2+2i)z_2=0$.

These both mean $z_1=-(1+i)z_2$.

So the eigenvectors are $\begin{bmatrix}-(1+i)z_2\\z_2\end{bmatrix}$; i.e., multiples of $\begin{bmatrix}-(1+i)\\1\end{bmatrix}$.

I leave determination of the eigenvectors for $\lambda_2=3+2i$ as an exercise.