I have the following problem:
Find a general solution of the system $x'(t)=Ax(t)$ for the given matrix $A$.
$A=\begin{bmatrix}9 & -6\\15 & -9\end{bmatrix}$
When I calculate the eigenvectors by hand I get $\pm\sqrt{171}$, but by Matlab I get these complex values of ridiculously small, complex fractions.
I can't figure out the disconnect.
My calculations are: $$(9-r)(-9-r)-6(15)=0$$ $$-81-9r+9r+r^2=0$$ $$r^2-171=0$$ $$r=\sqrt{171}$$
I'm not sure where I'm going wrong, since this section is supposed to be about complex solutions, but when I calculate it I don't get one and Matlab's doesn't seem right.
Your characteristic polynomial is $$\left|\begin{matrix} 9-r & -6\\ 15 & -(9+r)\\ \end{matrix}\right|=-(9-r)(9+r)-(-6)\cdot 15=-81+r^2+90=r^2+9,$$ so the eigenvalues (not eigenvectors) of $A$ are $r_1=3i$ and $r_2=-3i$.