I want to know how I can express a matrix exponential $e^{tA}$ for a matrix $tA$, in terms of real valued functions for a matrix with complex eigenvalues.
(Example, the initial value problem ODE $x'(t)=Ax(t), x(0)=a$, where $A$ is a constant matrix, has the solution $x(t)=e^{tA}a$, so the knowing how to calculate $e^{tA}$ helps gives the solution).
What I know:
The matrix $A=\begin{bmatrix} a & b \\ -b & a \end{bmatrix}$ has complex eigenvalues $\lambda_{1,2}= a \pm ib$, so we can express $e^{tA}$ as $$e^{tA}= \begin{bmatrix} e^{at} \cos(bt) & e^{at} \sin(bt) \\ -e^{at} \sin(bt) & e^{at} \cos(bt) \end{bmatrix}.$$ Which is good as all functions are real valued.
However what is the process for the general $n$ by $n$ matrix? There is the general formula for matrix exponential $$e^{tA}=\sum^{\infty}_{k=0} \frac{t^k}{k!}A^k=I=tA+\frac{t^2}{2!}A^2+\frac{t^3}{3!}A^3 +... $$ but if $A$ is not nilpotent matrix we get a series solution, which for obvious reasons I would prefer to avoid.
Additional edit based on comments:
- Example 1 How does this approach generalize to a $n$ by $n$ matrix? Why does $A=QAQ^{-1}$?
- Example 2 (look at example 4 in the document) How do I express a fundamental matrix in terms of real valued functions?