Linear System with Complex Eigenvalues: General Facts

DIFFEQ-N9O$7B

Which of the following statements is wrong about the linear system $x'=Ax$ where $A$ is real and has complex eigenvalues?

A

If $r=\lambda+\mu i$ is an eigenvalue of the matrix $A$ and $\xi=v_1+iv_2$ is a corresponding eigenvector, then, $\bar{r}=\lambda-\mu i$ is also an eigenvalue of $A$ and $\bar{\xi}=v_1-iv_2$ is an eigenvector for $\bar{r}$.

B

If $r=\lambda+\mu i$ is an eigenvalue of the matrix $A$ and $\xi=v_1+iv_2$ is a corresponding eigenvector, then $x_1(t)=Re(\xi e^{rt})$ and $x_2(t)=Im(\xi e^{rt})$ are two solutions of the linear system.

C

If $r=\lambda+\mu i$ is an eigenvalue of the matrix $A$ and $\xi=v_1+iv_2$ is a corresponding eigenvector, then, both $\xi e^{rt}$ and $\bar{\xi}e^{\bar{r}t}$ are solutions of the linear system. Further, their real and imaginary parts are also solutions of the linear system, which are real. Hence, we can get four independent real solutions of the original linear system.

D

If $A$ is of size $2\times 2$ and it has a non-real complex eigenvalue $r=\lambda+\mu i$, then we only need to compute the eigenvector of this eigenvalue to construct a fundamental matrix, without worrying about the other eigenvalue.