Express the general solution of x2 95 52 1x in terms of real

Express the general solution of x\'=(2, 9/5, -5/2, -1)x in terms of real-valued functions. (this is 2x2 matrix, 2 and 9/5 on the left, -5/2 and -1 on the right. I know that the roots are 1/2+3/2i and 1/2-3/2i, complex numbers. But how do I find a and b for the first root?)

Solution

The first eigenvalue is ?=1/2+3/2i. To find the corresponding eigenvector u=[a,b], substitute this into the eigenvalue equation, (A-?I)u=0. You get the following two equations:
(3/2-3/2i)a-5/2b=0
9/5a-(3/2+3/2i)b=0

You can now choose either a or b to be 1, and see what the other component has to be. If a=1, the first equation tells us
(3/2-3/2i)-5/2b=0, or b=3/5-3/5i.
We don\'t get any new information from the second equation. So the eigenvector is
u1=[1,3/5-3/5i]

Similarly, the second eigenvalue ?=1/2-3/2i gives the eigenvector
u2=[1,3/5+3/5i].