I'm getting a bit stuck on some algebra in the following problem. I'd appreciate suggestions on how to approach said algebra, and more generally, a method for solving dynamic systems with complex eigenvalues, as I tend to struggle with them.
In this case, s'pose we have a 2x2 matrix, representing a 2 dimensional initial value problem: $$X'=AX; X(0) = X_0 \quad A=\begin{bmatrix} a & -b \\ b & a \end{bmatrix}$$ It seems reasonable that we might use the matrix exponential identity, that $X=e^{At}X_0$ is a solution.
Very quickly, expanding small cases of $A^k$ gets a bit too hairy in the matrix multiplication for my liking. It can easily be seen that the eigenvalues of the matrix are going to be $a\pm bi$, with eigenvectors $[-bi\ b]$ and $[bi\ b]$. I'm getting stuck there though.
I believe the general form of the solution would be
$$ X\left(t\right)=c_{1}e^{\lambda_{1}t}\mathbf{x}_{1}+c_{2}e^{\lambda_{2}t}\mathbf{x}_{2} $$
where the $\lambda$ are the eigenvalues and $\mathbf{x}$ are the eigenvectors, and the $c$ are constants that depend on the initial conditions.
You might find the linear algebra lectures from Prof. Strang/MIT Open Courseware useful for additional details, particularly this lecture: https://ocw.mit.edu/courses/mathematics/18-06sc-linear-algebra-fall-2011/least-squares-determinants-and-eigenvalues/differential-equations-and-exp-at/.
I hope this helps.