1 real and 2 complex eigenvalues from

Question

1 Real and 2 Complex Eigenvalues   

From the real eigenvalue/vector pair, l₁ and ?h₁

e^l1t ?h₁

We find the other two solutions in identical manner which we did with the 2x2 case.  If the eigenvalues are l_2,3 = a + bI with eigenvectors ?h₂ and ?h₃   We can find two real-valued solutions through using Euler's formula to expand;

e^l2t ?h₂ = e^{(a + bi)t} ?h₂   = e^at (cos(bt) + i sin(bt))  ?h₂

in its real and imaginary parts, u? + i v? .  The final two real valued solutions we need are then,

u?                                           v?