I just had a quick question regarding linear autonomous systems of ODE with complex eigenvalues. in class we discussed how the solution can be written in the form $y(t)=e^{a t} QR(t)Q^{-1} y_{0}$ where $R(t)$ is the rotation matrix $\begin{pmatrix} \cos( bt) & \sin( bt)\\ -\sin( bt) & \cos( bt) \end{pmatrix}$ for some complex eigenvalues $\lambda = a\pm bi$ and Q is the matrix where each column consists of the real and imaginary parts of our eigenvectors $\mathbf{\vec{v}_1}= \mathbf{\vec{p}}+i\mathbf{\vec{q}}$ and $\mathbf{\vec{v}_2}= \mathbf{\vec{p}}-i\mathbf{\vec{q}}$.
Ive Tried to prove this formula using Euler's identity $e^{a\pm bi)t}=e^{at}(cosbt\pm isint)$ but I'm afraid im stumped, so if anyone has any diea how to prove this formula for our solution when we have complex eigenvalues I would really appreciate it
Thanks in advance!