System of first order differential equations

I'm having trouble solving this system of de's using eigenvalues and eigenvectors,

where the initial conditions are; x(0)=3 and y(0)=-4

I went through to calculate the eigenvalues and vectors to be;

and v1 = [1 , -i/2]

hence v2 = [1 , i/2]

Using this the general solution should be ?

__x__(t) = C1 . [1 , -i/2] + C2 . [1 , i/2]

where C1 & c2 are complex constants.

have i done this right so far?

To get the actual solution with the initial conditions i'm a bit confused can anyone help me here?

Many thanks.