nm = 2;
x = ToExpression["x" <> ToString[#]][t] & /@ Range[nm];
k = RandomComplex[2 + 2*I, {nm, nm}];
eqns = D[x[[#]], t, t] + Sum[k[[#, i]]*x[[i]], {i, 1, nm}] == 0 & /@
   Range[nm];
eigVals = Eigenvalues[k];
eigVecs = Eigenvectors[k];
eigVecsInverse = Inverse[eigVecs];
initsa = 0 == (Sum[
        eigVecsInverse[[#, i]]*D[x[[i]], t] /. t -> 0, {i, 1, nm}] +
       Sqrt[-eigVals[[#]]]*
        Sum[eigVecsInverse[[#, i]]*x[[i]] /. t -> 0, {i, 1, nm}]) & /@
    Range[nm];
initsb = Thread[(x /. t -> 0) == 1 + I];
sys = Join[eqns, initsa, initsb];
aSol = DSolve[sys, x, t];
TrigToExp[aSol]